ABS TRACTS. (15)

84. On the functional equation ƒÕn(x)=x, II. (in Japanese). Tsuruichi

HAYASHI. [Tokyo But. Z. 43 (1934), 151-153.] Some remarks on the literatures of

B abbage's equation are given. (S. K.)

85. On an oval. (in Japanese). Soji MATSUMURA. [Tokyo But. Z. 43 (1934),

153.] The following theorem is proved. In order that every chord joining opposite points

θ and θ+π of an oval p=p(θ) should be a double affine normal, it is necessary and suf

ficient that the radius of curvature of the oval should be a function of θ with the period π.

(S. K.)

86. On the summation Sp=nƒ°n=1 np (in Japanese). Masaji KADOWAKI.

[Tokyo But. Z. 43 (1934), 178-194.] The author finds an expression of Sp in a determinantal form. He also finds the relations between Sp and Bernoulli's numbers Bp, for example,

87. On a proof of Fermat's problem. (in Japanese). Yonoshin AIHARA.

[Tokyo But. Z. 43 (1934), 205-206.] The author finds the position of a point x, the sum of

the distances from which to the given three points ƒ¿, ƒÀ, ƒÁ becomes smallest, by a calculation

of complex numbers. (S. K.)

88. On the origin of the word Kika (geometry) and Daisu (algebra), II.

(in Japanese). Tsuruichi HAYASHI. [Tokyo But. Z. 43 (1934), 199-202.]

89. Kowa Seki and Calculus. (in Japanese). Yoshio MIKAMI. [Tokyo But .

Z., 43 (1934), 207-214.] The conclusion of the author's historical researches on the subject

is that the notion of differentiation can, by no means, be seen in the works of Kowa Seki,

and his treatises on calculus are confined to the definite integration. (S. K.)

90. Some numerical relations among the set of points, segments, surfaces,

etc. (in Japanese). Tamekichi HISHIDA. [Tokyo But. Z. 43 (1934), 214-221, 262-276,

303-312.] Many fundamental formulas of topology, similar to Euler's relation, are discussed

by an elementary method. The whole theory is established regarding the multiply connected

figures. (S . K.)

91. A problem of Kido Kobayashi. (in Japanese). Tsuruichi HAYASHI.

[Tokyo But. Z. 43 (1934), 247-251.]

92. Goethe and the natural science. (in Japanese) . Kunizo YONEYAMA.

[Tokyo But. Z. 43 (1934), 251-256.] Describing about Goethe, the author asserts that the

creations of arts and of science are performed in a same manner of mental activities.

(S. K.)

93. Two notes. (in Japanese). Soji MATSUMURA. [Tokyo But . Z. 43 (1934),

256-258.]

94. On a theorem of Minkowski. (in Japanese). Masaji ITO. [Tokyo But . Z.

43 (1934), 258-259.]

95. Anmei Aida and Kokogen (the geometry of triangles). (in Japanese) .

Yoshio MIKAMI. [Tokyo But. Z. 43 (1934), 297-302.]

96. On the deviation of normal of a plane curve. (in Japanese). Soji MA

TSUMURA. [Tokyo But. Z., 43 (1934), 302-303.]

97. On a property of nine point circle. (in Japanese) . Nobuo OSEKI. [Tokyo

B ut. Z. 43 (1934), 317-319.] The author proves the theorem: Let the circles O1, O2, O3

touch respectively the sides BC, CA, AB of a triangle ABC at their middle points and the

arc BC, CA, AB of the circumscribed circle O. Then the circle O4, which touches externally

O1, O2, O3, touches internally the inscribed circle and the nine-point circle of the triangle

at the same point. (S. K.)

(16) BSTRACTS.

98. On a theorem of Clifford. (in Japanese). Tadahiko KUBOTA. [Tokyo But. Z., 43 (1934), 337-338.] The author criticizes the memoir of Mr. W. B. Carver, "The failure of the Clifford chain", in the Amer. Journ. of Math., 1920. And he asserts that there is no failure in the theory of Clifford. (S. K.)

99. A geometrical problem on maximum. (in Japanese). Tadasu KAWAGUCHI. [Tokyo But. Z. 43 (1934), 339-344.] The author treats the problem to determine the positions of n points P1, P2, … …, Pn on a given circle, so that the sum S=ΣPkPlλ

(λ>0) becomes greatest. He succeeds to solve the problem, except the case 1<λ<2 and

the case 2<λ<λ1 n=odd, where λ1=(log2/3)÷log√3/2. (S. K.)

100. On geometry of circles and spheres. (in Japanese). Soji MATSUMURA.

[Tokyo But. Z. 43 (1934), 355-356.]

101. On geometry of a system of circles. (in Japanese). Soji MATSUMURA.

[Tokyo But. Z. 43 (1934), 396-400.]

102. Division table for three digits divisor. (in Japanese). Kiroku TOYO

KAWA. [Tokyo But. Z. 43 (1934), 404-410.] The method of using the table created by

the author is explained. (S. K.)

103. Enri (theory of circles) of Kowa Seki. (in Japanese). Tsuruichi HAYA

SHI. [Tokyo But. Z. 43 (1934), 427-432.] Some historical reports on the subject are given.

(S. K.)

104. On a theorem of Mr. Sawayama. (in Japanese). Yonoshin AIHARA.

[Tokyo But. Z. 43 (1934), 435-438.] A new proof is given to the following theorem of Mr.

Y. Sawayama. Let four points A, B, C, D lie on a circle O and let a circle touching O

touch AB and CD at P and Q respectively. Then the line PQ passes through the center of

of either inscribed or escribed circle of the triangle formed by any three of A, B, C, D.

(S. K.)

105. On the center of mean distance. (in Jappnese). Masaji KADOWAKI.

[Tokyo But. Z., 43 (1934), 463-466, 500-510.] The nature of the center of mass of weighted

points is applied to several geometrical problems on concurrency of lines and like others.

(S. K.)

106. Some properties of complete quadrilateral. (in Japanese). Yonoshin

AIHARA. [Tokyo But. Z. 43 (1934), 469-472.] The main theorem here obtained is as

follows: Let gi(i=1, 2, 3, 4) be the side of a complete quadrilateral, Aik be the intersection

of gi, gk, and Oi be the center of the circum-circle of the triangle formed by gk•Œs (k•‚i).

Then •¢A23A31A12•ä•¢O1O2O3. If the segments joining the corresponding vertices of these

two triangles are divided by the points M23, M31, M12 in a constant ratio, then •¢M23M31M12

is also similar to the preceding two triangles. Each two of these three triangles are in the

perspective positions, and those three perspective axes meet at a point. Especially, the axis

of •¢A23A31A12, •¢M23M31M12 touches the inscribed parabola of the quadrilateral. (S. K.)

107. Some remarkable straight lines. (in Japanese). Mikio KOBAYASHI.

[Tokyo But. Z. 43 (1934), 495-499.] The author considers a straight line L determined by

a triangle ABC and a point P, and he finds the curve enveloped by L, when •¢ABC moves

inscribing to a fixed circle and describing about a fixed conic. Similar problem is treated

in the case of a quadrilateral. Whole discussion is made by the calculation of complex

numbers. (S. K.)

108. On the deviation of a plane curve. (in Japanese). Soji MATSUMURA.

[Tokyo But. Z. 43 (1934), 499-500.] If the angle between the two normals, ordinary and

affine, at a point of a plane curve be denoted by φ, then tanφ=1/3dρ/dσ, where ρ is the

radius of curvature and dƒÐ is the differential of the curve length. Two other expressions of

ABSTRACTS (17)

tan φ are obtained, one of which is tanφ=1/3(dp/dθ+d4p/dθ4)dθ/dσ, where p=p(ƒÆ) is the polar

tangential equation of the curve. (S. K.)

109. Mathematical education and Edison. (in Japanese). Kunizo YONEYAMA. [J. Math. Ass. Jap. 16 (1934), 1-12.] The mental faculties of inventors and mathematicians are compared and criticized. The failure of mathematical education to an early age of Edison is pointed out. (S. K.)

110. A solution of biquadratic equation due to a Wasan expert. (in Japanese). Tsuruichi HAYASHI, [J. Math. Ass. Jap. 16 (1934), 12-17.] A Solution of Yasutaro Harada, a pupil of Chorin Kawakita, is written. It is quite similar to the solution

of Ferrari. (S. K.)111. A survey on the scientific investigations of the teaching of mathe

matics. (in Japanese). Shintaro NABESHIMA. [J. Math. Ass. Jap. 16 (1934), 17-23.]The meaning of the scientific investigations, its classification, actual examples and its criticism are described. (S. K.)

112. Formula for area of triangle and quadrilateral. (in Japanese). Satoshi FUJIEEA. [J. Math. Ass. Jap. 16 (1934), 23-24.] Numerous formulas (108 for triangle, 44 for quadrilateral) are collected, expressing the area from various deta of the triangle and quadrilateral. (S. K.)

113. On entrance examination and mathematical teaching. (in Japanese). Ryoichiro SATO. [J. Math. Ass. Jap. 16 (1934), 51-71.] Proper methods of the pre

paratory education are proposed. (S. K.)114. On teaching of geometrical problems. (in Japanese). Gen SUZUKI.

[J. Math. Ass. Jap. 16 (1934), 71-74.]115. On a problem concerning figures described on a fan. (in Japanese).

Torajiro YOSHITAKE. [J. Math. Ass. Jap. 16 (1934), 74-79.] A figure sketched on a surface of a fan reveals itself in a different shape when the fan is folded. The transformation between these two shapes is treated. (S. K.)

116. Tables for calculation of the roots of cubic equation with numerical coefficients. (in Japanese). Katsuzo YUASA. [J. Math. Ass. Jap. 16 (1934), 79-94.] Convenient tables are given and their use is explained, for calculation of the roots of a cubic equation by means of sinx, cosx, sinhx, coshx. (S. K.)

117. On some problems. (in Japanese). Soji NAKAJIMA. [J. Math. Ass. Jap. 16 (1934), 94-100.] Miscellaneous remarks are given, among which the followings are conspicuous.

1. The correlation coefficient

corresponds to the cosine of the angle between the two hyperspheres (xk), (yk).

2.

Here s+ƒ¢s and s+ƒ¢•Œs are the intersections of the curve and its chord, parallel to the

tangent at S. ƒÓ is the angle between the tangent and the affine normal at S. (S. K.)

118. On development of our association. (in Japanese). Toshichiyo IWA

MOTO. [J. Math. Ass. Jap. 16 (1934), 109-113.]

119. On errors of the functions N-1, •ãN, 3•ãN, m•ãN. (in Japanese). Hide

aki HIRANO. [J. Math. Ass. Jap. 16 (1934), 114-119.] The following facts are proved by

an elementary method, for the case k=-1, 1/2, 1/3, 1/m. If a positive number N is overesti

mated by x percent, then the error in excess for Nk is at most kx percent. (S. K.)

(18) ABSTRACTS

120. On a report of the commitee on individual differences. (in Japanese).

Ryoun TANAKA. [J. Math. Ass. Jap. 16 (1933), 119-125.]

121. On a certain multiplication table. (in Japanese). Zenzo KIKUTA. [J.

Math. Ass. Jap. 16 (1934), 125-133.] A convenient table for the multiplication of one of

1, 2, •c•c, 8, 9 and a natural number less than 10000 is given and explained. (S. K.)

122. A formula for ƒÎ and its extension, due to San Sakuma. (in Japanese).

Tsuruichi HAYASHI. [J. Math. Ass. Jap. 16 (1934), 133-139.]

123. A note on problem, 112. (in Japanese). Yonoshin AIHARA. [J. Math.

Ass. Jap. 16 (1934), 139-143.] Let the diagonals of a quadrilateral ABCD intersect or

thogonally at a point P. Draw PA•Œ, PB•Œ, PC•Œ, PD•Œ to the sides AB, BC, CD, DA respectively,

making a constant angle. Let the perpendicular to AB,•c•c, DA at A•Œ,•c•c, D•Œ meet the

opposite sides at A•••c•c, D•• respectively. Then the eight points A•Œ, A••,•c•c,D•Œ, D•• are

concyclic. This is an extension of the problem, No. 112, proposed in this journal. The author

proves this theorem and some other natures related to the figure. (S. K.)

124. On Hirano's method of finding square roots. (in Japanese). Masaji

TSUGAWA. [J. Math. Ass. Jap. 16 (1934), 143-156.] The author criticizes Hirano's

method, and shows some examples for which Hirano's method fails. (S. K.)

125. .Konigsberg's problem, treated from the point of view of mathe

matical teaching. (in Japanese). Kunizo YONEYAMA. [J. Math. Ass. Jap. 16 (1934),

167-178.] The author recommends to treat the universal problems as a good training in the

school mathematics. ( S. K.)

126. On various topics in teaching of mathematics. (in Japanese). Jiro

HAMADA. [J. Math. Ass. Jap. 16 (1934), 178-189.] The author observes about the

fundamental principles of mathematical education, which are, according to the author,

intuition, experience, unification, actual life, idea of function, utilization. (S. K.)

127. On methods of proof in geometry. (in Japanese). Jusiro NARAI. [J.

Math. Ass. Jap. 16 (1934), 190-197.] Several topics on the methods of proof are taken.

Above all, we see many suitable examples of geometrical problem, which are to be proved

by mathematical induction. (S. K.)

128. An impossible construction problem. (in Japanese). Tsuruichi HA

YASHI. [J. Math. Ass. Jap. 16 (1934), 198-199.] A circular arc BDC is drawn, outside of

a triangle ABC. The author takes the problem of dividing the area of the figure ARDC,

into equal areas, by a straight line passing through the given point D on the arc, and he

proves that the construction is impossible in elementary geometry. (S. K.)

129. On some problems in Japanese mathematics. (in Japanese). Senkichi

OKA. [J. Math. Ass. Jap. 16 (1934), 200-211.] Solutions of the several problems, treated

in old Japanese school of mathematics, are translated into the modern formulas. Among

them we see Horner's method for numerical equation, Guldin's formula for area and the

series denoting the periphery of an ellipse. (S. K.)

130. On analysis and synthesis. (in Japanese). Yoichi YOSHIDA. [J. Math.

Ass. Jap. 16 (1934), 241-247.] This is an address at the annual meeting of the Association.

Several meanings of the words "analysis" and "synthesis" are examined. (S. K.)

131. On extension of Simson's line to projective and affine geometry, I.

(in Japanese). Kotaro HIRANO. [J. Math. Ass. Jap. 16 (1934), 361-379.] This is translated from Mr. Takasu's memoir. Sur la generalisation projective et affine de la droite Simsonienne, Tohoku Sci. Rep. 20 (1931). (S. K.)

132. A machine for drawing conic sections. (in Japanese). Jun-ichi OYAMADA. [J. Math. Ass. Jap. 16 (1934), 387-392.] The illustration of the machine invented

by the author. (S. K.)133. On syllabus for synthetic treatment of girl's school mathematics.

(in Japanese). Ryusaku RITAMURA. [J. Math. Ass. Jap. 16 (1934), 393-408.] The

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author's view for the syllabus of the title. (S. K.)

134. On teaching of trigonoimetry in fundamental curriculum in middle

schools. (in Japanese). Yashiro KAMIBAYASHI. [J. Math. Ass. Jap. 18 (1934), 408-

416.] The author's view for the syllabus of trigonometry. (S. K.)

135. On the successive approximation to a root of an algebraic equation.

(in Japanese). Toshio UNO and Yoshiji HASHIMOTO. [J. Phys.-Math. Soc. 8 (1934),

249-253.] In the ordinary successive approximation to a root ƒ¿ of x=ƒÓ(x), the authors

transform previously the equation by a definite rule, so as the approximation to be alterna

tive. They also establish the following new method of approximation, in the case ƒÓ•Œ(0)=0.

(S. K.)

136. On the uniformly almost periodic functions of finitely or infinitely

m any variables. Tosio KITAGAWA. [Proc. Phys. Math. Soc., Ser. 3, 16 (1934), 39-

51.] The paper is aimed to deduce the fundamental theorems of the uniformly almost

periodic functions, introduced by Bochner, from the standpoint of a certain integral equation,

whose idea is based upon Weyl's method. (S . K.)

137. An extension of the vectorial domain of an oval for the relative

g eometry. Junko HIRASAWA. [Proc. Phys.-Math. Soc., Ser. 3, 16 (1934, 73-76.] Let

p(ƒÓ) be the function of supporting lines of an oval x, then p(ƒÓ) +p(ƒÓ+ƒÎ) is the function

of supporting lines of the vectorial domain X of x. Taking the relative distance

r(φ)=p(φ)/q(φ), (q(φ) is function for unit oval)

the author considers an extended vectorial domain X1 denoted by r(ƒÓ)+r(ƒÓ+ƒÎ), and

gets the inequalities analogous to those of Blashke, about the perimeters and areas of x and

X1. He also considers another extended domain X2. (S. K.)

138. On a continuity of function of many variables. Tosio KITAGAWA.

[Pros. Phys.-Math. Soc., Ser. 3, 16 (1934), 111-116.] The author proves the theorem: If

F(x1, x2,•c•c,xn) is a continuous monotone increasing function with respect to each of these

n variables for any fixed values of other variables in the vicinity of (x10, x20,•c•c, xn0), then

it is a continuous function of (x1, x2,•c•c, xn) in that vicinity. (S. K).

139. On some class of weighted means. Tosio KITAGAWA. [Pros. Phys.-

Math. Soc., Ser. 3, 16 (1934), 117-126.] The author deduces a class of means

of x1, x2,•c•c, xn, from some axioms about Mn. (S. K.)

140. Infinite series with multiply modotone terms. Shin-ichi IZUMI.

[Proc. Phys.-Math. Soc., Ser. 3, 16 (1934), 127-131.] Ostrowski's theorem on monotone

sequence is extended to r-ple monotone sequence, r being any number•†0, and the follow

ing theorem is proved and is applied to others. If (an) is a r-ple monotone zero sequence,

then ġan is convergent when and only when Ģr-1an exists and the sequence

σn(r)=Δn-r(Δr-1a0)-An(r)Δr-1an is convergent. Here

(S.K.)

141. On multiply monotone sequences. Diro ITO and Shin-ichi IZUMI.

(20) ABSTRACTS

[Pros. Phys.-Math. Soc., Ser. 3, 16 (1934), 132-136.] The discussion is made in the field similar to that of the preceding paper. Some natures of multiply monotone sequence itself

is proved. One of the main theorems is as follows. Let δ>0. Suppose that an=o(n-δ),

δan exists and (an) is r-ple monotone, for r>-δ. The series ∞ Σn=0Δ-δan, is convergent

when and only when Δ-1(Δran) exists and the sequence

is convergent. (S. K.)142. On a linear transformation of infinite sequences. Gen-ichiro SUNO

UCHI. [Proc. Phys.-Math. Soc., Ser. 3, 16 (1934), 161-163.] A kind of sufficient condition, quite different from that of Agnew (Tohoku Math. Journ., 35), is given, for that

should imply xn=o(1). (S. K.)

143. A generalization of the Carleman's inequality theorem. Gen-ichiro

SUNOUCHI and Naobumi TARAGI. [Proc. Phys.-Math Soc., Ser. 3, 16 (1934), 164-

166.] The following theorem is proved. Let av•†0, ƒÉv•„0, ƒ©n=ƒÉ1+ƒÉ2+•c•cƒÉn and

be convergent. Then there exists the inequality

where the constant e is best possible. (S. K.)

144. Uber die nach x partiell stetige Funktion f(x, y). Mituo INABA.

[Proc. Phys.-Math. Soc., Ser. 3, 16 (1934), 201-203, 225-226.] Let f(x, y) be a function

defined in a product space R1(x)R(y), where R1 is a linear space, and let it be continuous

for x and be of at most ƒ¿th class (of Baire) for y. Then f(x, y) is at most (ƒ¿+1) th class

with respect to (x, y). Even when ƒ¿ depends to x, it is sure that f(x, y) belongs to a certain

class of Baire. (S. K.)

145. On the Laguerre's theorem concerning the separation of real roots.

Itiro MURASE. [Proc. Phys.-Math. Soc., Ser. 3, 16 (1934), 204-208.] The author proves

Laguerre's theorem by an elementary method of algebra, and extends it to a wider range,

which includes the theorem of Fourier. (S. K.)

146. On a class of meromorphic functions. Kosaku YOSHIDA. [Proc.

Phys.-Math. Soc., Ser. 3, 16 (1934), 227-235.] Consider the following properties of function

y(z).

(A) For any given sequence {ai}, {y(z+ai)} is a normal family in any closed and

finite domain.

(B) {y(z+ai)} has no partial sequence having constant as a limiting function.The class of meromorphic functions having the property (A) (class 4(A)) and the

class having the properties (A) and (B) (1st category) are concerned. The author proves

that: I. In order that y(z) may belong to the class (A), it is necessary and sufficient that

|y•Œ(z)|{1+|y(z)|2}-1 should be limitd. II. A function y(z) of 1st category has no as

ymptotic values and is of order 2 such that lim inf N1 (r)T-1(ry)=2. (S. K.)

147. The absolute summability (A) of the conjugate Fourier series.

Tatsuo TASAHASHI. [Proc. Phys.-Math. Sac., Ser. 3, 16 (1934), 236-243.] A sufficient

condition for the absolute summability (A) of the conjugate Fourier series of f(x) is found

in terms of

ABSTRACTS (21)

Also a condition is obtained, for that the absolute summability implies the ordinary convergence. (S . K.)

148. Sur le faisceau canonique. Yasuo MORI. [Proc . Phys.-Math. Soc., Ser. 3, 16 (1934), 265-267.] The author introduces a new kind of canonical family of lines with respect to a surface, which has the same properties as the family considered by Sannia , but the simpler forms. (S. K.)

149. Sur la serie d'interpolation de Stirling. Tosio UNO et Yosiharu HASIMOTO. [Proc. Phys.-Math. Soc., Ser. 3, 16 (1934) , 268-272.] The authors prove that the interpolation series of Stirling

converges to the same finite limit or diverges to •‡, according as the series

converges or diverges to •‡. (S . K.)

150. A few self-reciprocal functions. Brij Mohan MEHROTRA . [Proc.

Phys.-Math. Soc., Ser. 3, 16 (1934), 273-274.] The function f(x) satisfying the equation

is said to be +Rv, or -Rv according as ƒÃ is -1 or +1. The author finds that

is •}Rv or •‚Rv according as f(x) is +RV or -Rv. (S . K.)

151. On Cesaro's theorem. Shin-ichi , IZUMI and Gen-ichiro SUNOUCHI.

[Proc. Phys.-Math. Soc., Ser. 3, 16 (1934), 297-302.] Gesaro's theorem on the distribution

of positive and negative terms of a conditionally convergent series is extended in two ways,

once in Rademacher's sense and then in consideration of r-ply monotone sequence. The

following theorem is added. If pn is the number of the cifer 1 in the first a cifers of the

diadic fraction of x(0<x<1), then

for almost all x in (0, 1). (S . K.)

152. Vierscheitelsatz im konformen Raume. Tsurusaburo TAKASU . [Proc.

Phys.-Math. Soc., Ser. 3, 16 (1934), 303-313.] The points on a space curve, in which the

corresponding osculating circle contains at least four consecutive points, are called "Schei

tel". The author proves the theorem: The curves of a certain class of closed space curves

have 2n "Scheitel", where n=2, 3, 4,•c•c. For the proof he follows the example of Herglotz

(W. Blaschke, Vorl. ub. Dif. Geom., I, 3 Aufl. p. 31) for the ordinary "Vierscheitelsatz" by

modifying his conformal-geometrical generalization (T. Takasu, Differentialkugelgometrie, I,

Tohoku. Sci. Rep., 17 (1928), p. 289) of the Serret-Frenet formulas. Author .

153. On the homothetic figures. Masanori YAMANOUTI . [Proc. Phys.-

Math. Soc., Ser. 3, 16 (1934), 314-317.] The following theorem, which was once proved by

Mr. Nakajima (Matsumura at present) in the case n=3, is proved generally. If there exist

two bodies in n-dimensional space, whose projections to any hyperplane of n-1 dimensions

are homothetic to each other, then the bodies themselves are homothetic. (S, K.)

154. On the linear displacement in the (generalized manifold. Kentaro

(22) ABSTRACTS

YANO. [Proc. Phys.-Math. Soc., Ser. 3, 16 (1934), 318-326.] E. Cartan has introduced a

plane element associated to every point in the metric manifold. In this paper, the theory

of linear displacement is developed in such a manifold Kn. The tensor calculus in Kn is

established and especially the curvature tensors are otained. (S. K.)

155. On some inequalities. Katsutaro KOBAYASHI. [Proc. Phys.-Math.

Soc., Ser. 3, 16 (1934), 341-344.] A main theorem of the paper is as follows, and it is an

extension of Jensen's inequality.

Let k•†0, 0<p<q, av•†0. Then

according as k> or <1, where the equality sign is permissible when and only when

a1=a2=•c•c=an. (S. K.)

156. On some inequalities. Shin-ichi IZUMI, Katsutaro KOBAYASHI and

Tatsuo TAKAHASHI. [Pros. Phys.-Math. Soc., Ser. 3, 16 (1934), 345-351.] The preced

ing inequality of Kobayashi is extended. Let ƒÉ>0, av•†0 and ƒÅ=ƒÆ (ƒÉ, x) be a positive,

strictly increasing and differentiable function of x(•†0) and of ƒÉ, for which we denote

x=Į-(ā;ă). Then

is the monotone decreasing function of A provided that k>1 and logĮ (ă, x) is convex with

respect to ă. (S. K.)

157. Another proof of the inequalities concerning the vectorial domain

of the oval and ovaloid, and its extension for the relative differential geo

metry. Junko HIRAKAWA. [Proc. Phys.-Math. Soc., Ser. 3, 16 (1934), 352-350.] The

known inequalities concerning vectorial domain are reproved by a method of the relative

differential geometry. His extensions of the inequalities, obtained in this proceedings p. 73

are corrected here in some points. (S. K.)

158. Thorems on limits of recurrent sequences. Gen-ichiro SUNOUCHI.

[Proc. Imp. Acad. 10 (1934), 4-7.] The author reproves the theorems due to Copson, Ferrar

and Walsh. Finally he gets the extended theorem as follows. Let

diverge, and

converge.

Then

implies tn=0(1). (S. K.)

159. A theorem on Cesaro summability. Tatsuo TAHAHASHI. [Pros.

Imp. Acad. 10 (1934), 8-9.] Extending Okada's theorem, the author proves that the series

Σan is summable (C,k) with sum s, provided that Σan is summable by Abel's method

with sum s and

(S. K.)

160. The foundation of the theory of displacements, II (Application to the function manifolds). Akitsugu KAWAGUCHI. [Proc. Imp. Acad. 10 (1934), 45-48.] By specialization of the general theory set out in the previous paper (this Proc. 9), all kinds of the displacements of Kawaguchi, Michal and Moisil in the function manifolds

ABSTRACTS (23)

is discussed systematically. Vitali's theory in the Hilbertian space is also a special case and moreover the author deduces the displacement of Wundheiler and Hlavaty from his theory by specialization. Author.

161. Kinematic connections and their application to physics. Toyomon HOSOKAWA. [Proc. Imp. Acad. 10 (1934), 49-52.] Recently a new physical theory has been developed by O. Veblen, J. A. Sbhouten and others in which the principal point is funded on a projective connection. in the present paper we shall develop some connections in the manifold admitting the kinematic transformations, and shall give a unification of the gravitational field not only with the electromagnetic, but also with Dirac's theory of material waves. Author.

162. On the convergence factor of the Fourier-Denjoy series. Fu Traing WANG. [Proc. Imp. Acad. 10 (1934), 53-56.] The object of the paper is to show that n-1 is a convergence factor of Fourier-Denjoy series

namely the series Σ1/n (an cosnx+bnsinnx) converges almost everywhere, and to construct

an example for which n-ƒÂ(0•ƒƒÂ•ƒ1) is not a convergence factor. (S. K.)

163. Einige Eigenschaften der polaren Kongruenzen in der projektiven

Flachentheorie, I, II, III. Yasuo MORI. [Pros. Imp. Acad. 10 (1934), 59-61, 62-64,

311-314.] Consider a rectilinear congruence H1, whose generating line l1 passes through

a point x upon a non-ruled surface (x). The polar congruence H2 of H1 is generated by

the line l2, which is conjugate to l1 with respect to any quadric of Darboux at x of the surface.

For brevity the harmonic conjugate point P of x with respect to two focal points of

the line l1 will be called the Konigs' point of the congruence H1 at x . Correlatively in the

plane pencil whose axis is l2 the harmonic conjugate plane ƒÎ of the tangent plane at x

with respect to two focal planes of the congruence H2 will be called the Konigs' plane of

the congruence H2 corresponding to the point x .

The point P is the pole of the plane ƒÎ with respect to the quadric of Lie .

The author proves that the envelope of the Konigs' planes of all the flex-ray congru

ences H2 of Lane corresponding to x is the Wilczynski-Bompiani quadric, and correlatively

the locus of the Konigs' points of all the cusp-axis congruences H1 of Lane at x is also a

certain quadric of Darboux correlative to the aforesaid quadric.

In the third paper the author proves that the locus of the Konigs' points of all the

canonical lines at P is a conic which has the contact of the second order at P with the

surface, and certain of its relations with the quadrics of Darboux at x are treated.

Author.

164. A new concept of integrals, II. Shin-ichi IZUMI. [Proc . Imp. Acad.

10 (1934), 57-58.] This continues the paper with the same title, same Proc. 9, 570-573,

which is abstracted in No. 276 of the preceding volume. (S . K.)

165. The foundation of the theory of displacements, III (Application

to a manifold of matrices). Akitsugu KAWAGUCHI. [Proc. Imp. Acad. 10 (1934), 13

3-136.] From the standpoint of the general theory set out by the present author, the

three kinds of displacements are introduced into a manifold of matrices, which may be

regarded of indeterminate dimensions. After difining the parallel system, the curvature

matrices and so on, the author applies the theory to an ennuple in a manifold with a linear

connection and to the linear transformation groups. Author.

166. Ein Beweis fur Szegosche Vermutung uber schlichte Potenzreihen.

Kenzo JOH und Shin-ichi TAKAHASHI. [Proc . Imp. Acad. 10 (1934), 137-139.]

Szego's imagined theorem is completely proved, in the following form. If

fk(x)=x+ak+1xk+1+a2k+1x2k+1+•c•c

(24) ABSTRACTS

is regular and schlicht for |x|＜1, then an=0 (n-1+2/k) where n=vk+1. (S. K.)167. A new proof of the Andersen's theorem. Shin-chi IZUMI. [Proc.

Imp. Acad. 10 (1934), 140-142.]

168. Ein Kriterium, fur normale einfache hyperkomplexe Systeme.

Kenjiro SHODA. [Pros. Imp. Acad. 10 (1934), 195-197.] In dieser Note wird bewiesen:

Eine Algebravom Rang n ist dann und nur dann normal-einfach, wenn der Modul

•~ mit einem allgemeinen Element x den Rang n2 hat. Daraus folgt leicht auch ein

Determinantenkriterium. Verfasser.

169. Ein Satz uber p-adische Schiefkorper. Tadasi NAKAMURA. [Proc.

Imp. Acad. 10 (1934), 198-199.] The following fact is proved. Es seiein -adische

Schiefkorper vom endlichen Rang uber seiem Zentrum K,ein Unterschiefkorper von,

der K enthalt. Die Gesamtheit der mitelementweise vertauschbaren Elemente von

bildet einen Unterschiefkorper. Sind<•Œ zwei Unterschiefkorper von, so ist die

Verzweigungsordnung bzw. der Restklassengrad zwischenund•Œ gleich dem Restklassengrad

bzw. der Verzweigungsordnung zwischenand•Œ. (S. K.)

170. On the convergence factor of Fourier-Lebesgue series. Fu Traing

WANG. [Proc. Imp. Acad. 10 (1934), 299-302.] Let

The author proves that 1) If |f(t)|ｐ(p>1) is summable and ∫0t φ(t)dt=o(t), then

is convergent for t=x. 2) If |f(t)|p(p≧1) is summable, then

Σn (ancosnt+bnsinnt)

is convergent for all t and ƒÂ>0. (S. K.)

171. Die naturliche Gleichung der Minimalkurven im konformen

Raume. Tsurusaburo TAKASU. [Proc. Imp. Acad. 10 (1934), 303-306.] A conformal

- geometrical generalization of the Study's theory (E. Study, Trans. Amer. Math. Soc., 11

(1910), p. 259) on the natural equation of the minimal curves in the Euclidean space.

"Berichtigung" as a slip in: Proc . Imp. Acad., 10, No. 3 (Oct. 1934). Author.

172. Remarks to some theorem of Burnside. Masatada TAZAWA. [Proc.

Imp. Acad. 10 (1934), 307-310.] A purely algebraic proof is given to the theorems of

Burnside, concerning the representations of group, which were proved by him using a power

series. (S. K.)

173. Diskriminantensatz fur normale einfache hyperkomplexe Systeme.

Kenjiro SHODA. [Proc. Imp. Acad. 10 (1934), 315-317.] Die beim Kriterium der nor

malen einfachen Algebren auftretende Determinante kann man zur Definition des Diskri

minantenideals im Kleinen und, als Produkt solcher Ideale, auch im Grossen verwenden. Fur

solcher Ideal wird der Diskriminantensatz bewiesen: Ein Ideal des Grundkorpers its dann

u nd nur dann durch das Quadrat eines Primideals der Algebra teilbar, wenn es in das

Diskriminantenideal aufgeht. Verfasser.

174. Diskriminantenformel fur normale einfache hyperkomplexe Sys

teme. Kenjiro SHODA. [Proc. Imp. Acad. 10 (1934), 318-321.] In dieser Note wird die

explicite Formel fur das Diskriminantenideal einer normalen einfachen Algebra bestimmt.

Vgl. die Berichtigung am Schluss der nachfolgenden Note von K. Shoda und T. Nakamura.

Verfasser.

ABSTRACTS (25)

175. On the Wiener's formula. Shin-ichi IZUMI. [Proc. Imp . Acad. 10 (1934), 393-398.] Suppose that the limit

exists and is finite. The author obtains the conditions for the validity of the formula

one of which is an extension of Bochner's condition. (S . K.)

176. Uber die monomial darstellbaren endlichen Substitutionsgruppen.

Masatada TAZAWA. [Proc. Imp. Acad. 10 (1934), 397-398.] The monomial representation

of a substitution group, whose order is a product of different primes, is always possible . The

author considers a general case of order, and obtains a condition for the possibility of

representation. (S. K.)

177. Uber das Produkt zweier Algebrenklassen mit zueinander primen

Diskriminanten. Kenjiro SHODA und Tadasi NAKAMURA. [Proc. Imp. Acad.

10 (1934), 443-446.] Die in der Theorie der galoisschen Zahlkorper bekannten Satze uber

die Diskriminante des zusammengesetzten Korpers werden in dieser Note auf die Theorie

der Algebren ubergetragen. Verfasser .

178. Uber die Definition der Shodaschen Diskriminante eines normalen

einfachen hyperkomplexen Systems. Tadasi NAKAMURA. [Proc. Imp. Acad.

10 (1934), 447-449.] The author asserts that Shoda's definition of the discriminant of a

normal, simple, hypercomplex system is given im Kleinen, and he gives a direct definition

im Grossen of the discriminant. (S. K.)

179. Some remarks on a theorem concerning star-shaped representation

of an analytic function. Kenjiro NABETANI. [Proc. Imp. Acad. 10 (1934), 537-

540.] The following theorems are proved. The radius of the greatest circle with the centre

at the origin which is represented conformally by every function f(z)=z+a2z2+•c•c(|z|•ƒ1)

on a star-shaped domain with respect to the origin under the condition

(1)

is given by the positive root of R2/

(1-R2)3=

1/

M2-1.

And under the same condition (1), f(z) represents |z|•ƒR on the convex domain where R

is given by (M.T.)

180. Note on the system of the orthogonal functions. Tatsuo TAKAHA

S HI. [proc, Imp. Acad. 10 (1934), 541-543.] Let {ƒÓi(s)} be an orthogonal system of

functions which are defined and squarely integrable in (0, 1) and x(s) be a function defined

in the same interval. The formal series

∞Σi=1φi(s)∫01x(s)φ(s)ds

is called the expansion of x(s) by the system {ƒÓi(s)}. Concerning the expansion (1) H.

Haar had, in his dissertation, proved the following theorems.

I. Let s0 be a point in (0, 1) and put

Kn(s0,s)=nΣi=1φi(s0)φi(s)

(26) ABSTRACTS

and

If {ƒÖn} is not bounded, then there exists a continuous function whose expansion diverges

at s=s0.

II. If every continuous function is uniformly approximated by the system {ƒÖi(s)} in

(0, 1) and {ƒÖn} is bounded, then the expansion of every continous function converges at

s=s0. In this paper the author proved these theorems by using theorems in the theory of

linear operation and at the same time proved to the theorem.

III. If the hypothesis I is satisfied, then the set of functions whose expansions are

divergent (indeed the partial sums are not bounded) at s=s0, is of the second category in

the space of all continuous functions. (M. T.)

181. Note on a curtain multivalent function. Tetuzi ITIHARA. [Proc.

Imp. Acad. 10 (1934), 544-545.] The following theorem is proved. Let

be regular and k-valent in 0＜|z|＜1, then

k|ak|2+(k+1)|ak+1|2+(k+2)|ak+2|2+… … ≦k. (M.T.)

182. Sur la theorie des ensembles analytiques dans les espaces abstra

its. Kinjiro KUNUGUI. [Pros. Imp. Acad. 10 (1934), 540-549.] The closure (fermeture)

E of a set E is, as usual, E E+E•Œ and a space (v), where every closure is closed, is called

a quasi-accessible space. Then the author proved the following theorems. (I) (Steinbach)

R un espace quasi-accessible et S un espace metrique complet separable. In projection P

d'un ensemble analytique M dins l'espace R•~S projete sur R est un ensemble analytique

dans R. (II) Soient R un espace quasi-accessible et S un espace metrique complet separable.

La projection P d'un ensemble d'unicite M dans l'espace R•~M projete sur R est un ensemble

d'unicite dans R, si pour tout point b de R, l'ensemble M•E(b•~S) est au plus d'un point.

(III) Soient R un espace quasi-accessible et S un espace de tous les nombres reels 0•…s•…1.

Tout ensemble dans R crible au moyen d'un crible analytique dins R•~S est un ensemble

analytuque. (IV) Soit R un espace quasi-accessible ou tout ensemble ouvert est un ensemble

analytique. Pour deux ensembles analytiques quelconques A et B, il existe duex ensembles

complementaires analytiques D et H tels que A-B •ºD; B-A•ºH; D• H=0. (V) Soit

F une famille d'ensembles telle que FC•ºFA. Alors pour deux ensembles A et B quelconques

de FA, il existe deux ensembles D et H de FAC tels que A-B•ºD; B-A•ºH; D•EH=0.

(VI) Soient R un espace quasi-accessible et Si, S2,•c•c, Sn des espaces de tour les nombres

reels. Si l'equation f(x, y1, y2,•c•c, yn)=0, x•¸RiyiSi i=1,•c•cn, ou f(x,y1, y2•c•cyn) est

une fonctionelle d'unicite, admet une solution y=yi(x) uniformes pour chaque point du

domaine d'existence, ce domaine d'existence est necessairement un ensemble d'unicite et les

solutions y=yi(x) sont des fonctionnelles d'unicite. (M. T.)

183. Uber die Verteilung der Primzahlen. Heihachiro ISHIKAWA. [Sci.

Rep. Tokyo Univ. Lit. a. Sci., Sec. A, 2 (1934), 27-40.] The following three theorems are

proved, among which (3) can be easily deduced from (1) and (2).

(1) pr sei die rte Primzahl, so besteht

pn+pn+1≧pn+2;

das Gleichheitszeichen tritt nur dann ein, wenn n=1 its.

(2) π(x) sei die Anzahl der Primzahlen≦x, so its fur x, y≧5

π(xy)>π(x)+π(y).

(3) pa pb>pab und{nΠr=1pr}2/n >pn+1. (S. K.)

ABSTRACTS (27)

184. Some remarks on the univalency and multivalency of functions.

Shigeo OZAKI. [Sci. Rep. Tokyo Univ. Lit. a. Sci., Sec. A, 2 (1934), 41-55.] The author

considers a meromorphic function of the form f(z)=1/z+a0+a1z+•c•ctogether with a

regular function of the form ƒÓ(z)=z+b2z2+•c•c, both defined in the domain |z|•ƒr.

Many theorems on the univaleney are discussed in a parallel manner for these two types of

the the functions. For example:

If ak•†0 (k=1, 2,•c•c), then the necessary and sufficient condition for the univalency

of f(z) is that

1≧|a1|r2+2|a2|r3+… …+n|an|rn-1+… ….

And if bk•…0 (k=1, 2,•c•c), then the necessary and sufficient condition for the univalency

of ƒÓ(z) is that

1≧2|b2|r+3|b3|r2+… …+n|bn|rn-1+… ….

Many other theorems concerning the multivalency of the function of the the forms

and φ(z)=zk+bk+1zk+1+… …

are obtained, in connection with 2F′(z)/

(z) a nd

zΦ ′(z)

/Φ(z).(S. K.)

185. On the multivalency of functions. Shigeo OZAKI. [ci. Rep. Tokyo

Univ. Lit. a. Sci., Sec. A, 2 (1934), 99-102.] The fundamental conception of the paper is

that, if f(z) is regular and k-valent in D, the total variation of amp df(z) on the whole

boundary of D is not less than 2kƒÎ. The following theorem is one of the consequences of

this idea.

Let f(z) be regular for |z|•…1 and

dθ ＜2(k+1)π for z=eiθ.

Then f(z) is at most k-valent in |z|•…1. (S. K.)

186. On absolute summability (A) of allied series and derived series of

urier series. Motoji KUNIYEDA. [Sci. Rep. Tokyo Univ. Lit. a. Sci., Sec. A, 2

(1934), 103-113.] Let

(1)

Extending Prasad's theorem, the author proves that the allied series of (1) is absolutely

summable (A), if ƒÆa(t) is of bounded variation in (0,ƒÅ) for some ƒ¿•†0. He also proves

that the derived series of (1) is absolutely summable (A), if the integral•ç0ƒÅ|ƒÆƒ¿•Œ(t)|/t at

exists for some α ≧0. (S. K.)

187. Zur Theorie (der gewohnlichen Differentialgleichungen. Hidegoro NAKANO. [Journ. Fac. Sci. Tokyo, Sec. I, 3 (1934), 1-63.] Verfasser definiert eine normale Differentialkette, namlich ein System von Differentialgleichungen

(28) STRACTS

mit den Bedingungen

Wenn eine gewohnliche Differentialgleichung eine Losung besitzt, welche mit der Differentialgleichung durch eine normale Differentialkette sich verbinden lasst, so heisst diese Losung eine eigentliche Losung. In dieser Abhandlung versucht Verfasser die Eigenschaften einer eingentlichen Losung, und beschreibt im Falle von linearen Differentialgleichungen eine Erweiterung des Sturmschen Komparationssatzes. Verfasser.

188. Zu den Abbildungen durch analytische Funktionen mehrerer kom

plexer Veranderlichen. Hidegoro NAKANO. [This journ. 11, 1-8.]189. Uber die Matrixfunktion. Hidegoro NAKANO. [This journal 11, 9-14.]190. Ein Beitrag zur Differentialgeometrie der zweidimensionalen Man

nigfaltigkeit im Raume von vier Dimensionen. Tadahiko KUBOTA. [This journal 11, 15-20.]

191. Uber die endlichen Gruppen der Algebrenklassen mit einem Zerfall ungskorper. Kenjiro SHODA. [This journal 11, 21-30.]

192. Einige Bemerkungen zur Affinflachentheorie. Tadahiko KUBOTA. [This journall 11, 31-34.]

193. Ideale in quadratischen Zahlringen. Yasuo AKIZUKI und Yoshiro MOBI. [This journal 11, 35-58.]

194. Zur Theorie der gewohnlichen linearen Differentialgleichungen. Hidegoro NAKANO. [This journal 11, 59-130.]

195. Konstruktionen gewisser algebraischer Zahlkorper durch die Modulfunktionen zweier Variablen. Teil I. Masao SUGAWARA. [This jounal 11, 131-184.]

196. Sur la geometrie projective des espaces a connexion affine. Sigekatu WATANABE. [This journal 11, 185-194.]

197. Ein neues Dualitatsprinzip bei den Kurven im konformen Raumen. Tsurusaburo TAKASU. [This journal 11, 195-212.]

198. On the strong summability of Fourier series. Tatsuo TAKAHASHI.

[This journal 11, 213-222.]199. Eine Bemerkung zum Blaschke-Howeschen Satze uber die Tan

genten einer ebenen algebraischen Kurve. Tadahiko KUBOTA. [This journal 11, 223-224.]

200. Uber die Theorie der Kreiskorper der lv-ten Einheitswurzeln, II. Taro MORISHIMA. [This journal 11, 225-240]

201. Uber die Fermatsche Vermutung, XI. Taro MORISHIMA. [This journal 11, 241-252.]

202. Une generalisation du theorme de Aiyar et du theoreme de Bricard. Tadahiko KUBOTA. [This journal 11, 253-254.]