//;■ ■■..Λ..,

DE

DERIVATA »:ti ORDINIS

FUNCTIONIS CUJUSLIBET m QUANTITATUMCOMMENTATIO

QUAM

EX SPECIALI REGIS GRATIA

ET CONSENSÜ AMPLISS. FACULT. PHILOS. UPSAL.

p. p.

JAGOBUS NICOL GRANLUNDPHIL. CAND. ST1P. NOKBERG.

ET

OLAVUS AÜGÜSTÜS ÖHSIANHED. PHIL. CAND.

NORRLANDI.

IN AUDITORIO GUSTAVIANO DIE VI MAJI MDCCCXLVIII.

Η. Ρ. Μ. β.

Ρ. IV.

UPSALIAEWAHLSTRÖM ET C.

<»)'

25

'\dy,<> ) Λ dy,'* )

\ dy,'-' )'

V dy,> /

■»Jff, + r_1K2 + ... + '-'Kn-i

r— 1 r ! Γ— 1+ r~ L2 + ... + T~ La— +r~1Ln

id^_yr\B> + TB 2 (d%\r«i + T'{* + + '«-i\dxnj \dxn~l J \dx*JIdy \% + Γ·ί-2 + ... + rLn— ι + TLuidx / '

ubi, brevitatis caussa positis:*At = a1«!... *at j

*1?! = a1«! alBbl ?

"i?2 = a xal.. .s_1 bl sb2 + a lay.. .8_1 s62 + ....+b lb2 ..*~lb2 *b2,

(2)<aKl—a1a1...s'la1Bkl9*K2 = a la2.. .s lSÄ2 + a '«2.. .*~lb2Bk2 +· · · ·+δ 1 &2· · ^8*1A2SA2,./tn_l == Λ'ftA . . ·* ^ ^n-l+^ 'ßj < · .S '^2 ...S '^2 ^b-1+·

+ a l(lv'·s Vvi B^n-i+ · · · · + Α Άα.|...8-1An_i SA„.1 ?

t=3 Ö . . . β~1Λ1 8/χ y

BL2 z= αια^. ayaL.. .smib2sl2 +.... +blb2... s-lb2

4

26

°Ln_i = a lat ...a-ikl%.l+a}al...e-ikiBla_i+....+b % ...8lA2s/n_i+( + rt1«1..«8"1An_1sZn_1 + + klkn_t....e ika,isln.i 5

(2) jsZ,n=a ...13ΐΖ2 sZn+α 'a,... Ll +... + b lb,... slZ2s/n +ι ....+ala1 ...8-1Zn_lsZn+...+A1 An_t...8",Zn_1sZn + alal...s Zn Zn+....1

.... + Z,Zn...e-1/nsZn*),

safcisfaciendum erit conditionibus:

(itf) Mß+ (tt-i) b + .... + 2A + Ζ =M) Z= ß-hb + ·.. + A + Z?

j (1l!i1)n,ß1 + (n-i),fc1 + .... + 2,i1+1Z11 ,£1 = ,λ1 + 1Ζ>1 + ...4· ,/ϊ1 + 1ΖΙ 9

(ii-l)'Z»2+.... + 2,A2+1Z2 =n-l5 ,Z2=1Z»2+... + 1A2+1Z29

P)(](1 ) 21ka_i+=2, V, = "A*.* + Vi ,l(lMn)Hn=i,

*) Satis perspicuum esse putamus, quomodo mquationes (2) sint intelligend®.Nam si in dextro membro oequationis cujuslibet factorem quemlibet termini cujus-libet consideramus, apparet, indice superiore significari numerum factorum ante-cedentium, indice vero inferiore ordinem factoris seu litter® antecédentis. Quumautem numerus litterarum a, b,... k, l sit = η (exceptis tantum litteris, qu®in paragrapbo 1 indicem 1 sinistra parte babent), sequi tur, esse a litteram pri-mam, b secundam, ...k (n-l):am, l n:tam. Admonemus etiam, quod c condi¬tionibus, quibus subjiciend® sunt b® lifter®, patet, litterarum, quarum indexinferior sit ff, tf-l primas deesse. Itaqqe litterarum, quarum index inferior est I,nulla drest, restant omnes litterarum, quarum index inferior est 2,deest a2 , restant b2 ,... Zr2 , Z2 ; litterarum, quarum index inferiorjest η—1,desunt η—2 prim® ßn—1 , Ζ^—χ ·> · · ·» restant du® ka_x et Zn_1 ; litterarum ,

quarum index inferior est », desunt an, bn,.../cn, restat Zn,

27

('-•Μ,)η'-ι«,+(((-1)'-Ά,+.... + 2'-Ά1+Γ-νι =η,! '"'ti = r-ie,+'-'6i+"" +'"'Äj?

[ ("'iMi) (n-1)'-'+2=«-1,

I ^•■'Κ.ι+-%.ι=2, '-'C. ='-τ*„-,) ('-■>/„) -<„ = 1, -t„ =Ρ)( (Ήι) n'«1+(n-i)'61+....+2'A1+'/i =η,

Ι rij = 'ilj +Γ61 + ···· + Γλ1 +Γ^! 5

Ι (τΜ2) (η-i)r6ä + .... + 2rA2 + Γ/2 = η-I,Ι r£a = r&2 +..., + rAa + r/2,

| m_l) ^ Γ^η—1 + ^a—1 — ^ ? Γίη—1 — ^n-l + ^n—1 7W r/„ = i, r4 = rin.

Demonstratio. Quoniam haec propositio theorematibus II,III convenit, si r = l, r = 2; restat, ut demonstremus, sivera ponatur baec propositio, veram etiam fore propositio-nem, quae inde, substituto r + 1 pro r, orietur. Quam pro-positionem ita exprimamus:

Si u = f(y) est functio quaelibet quantitatis variabilis*/?2/ = 9>(2Λ) functio quaelibet quantitatis variabilis 2bj«/x=91(^2)functio quaelibet quantitatis variabilis y2 9i...y =y>r(yt. + t)functio quaelibet quantitatis variabilis yr + ii et *Jr +

functio quaelibet quantitatis variabilis independentis χ: eritderivata n:ti ordinis:

28

η Γ(η+1) 1J [Γ(η+1)]Γ+ [r(w)]r+ Χ + Γ+ X

r + 1 «■ 4- Γ + 1 Κ· > ,r + lr*[ Γ (3)] Kl+ **+·" +

rWoV,r + 1ίΙ + Γ+1ιί+... + Γ+,ι»-ι + Γ + ,ιη[r(2)J

γ(λ+ι) r(b+1).... r(k+i)r(l+1)1 1

[/·(■«,+ 1)1« [Γ('«,1+1)]«[/'('4,+1)Ρ1

""ΪΓ(Ά1+1)]»[Γ(Άί+1)]!·....[Γ(Άη_ι+1)]'= '1

'

[Γ(,;1+ι)!«[Γ('ίί+ι)ρ....[Γ(,ζ_1+ΐ)|'.[Γ(7η+ΐ)ΐ''

ι ι

[r(a, + iy\"'A<

[r(k+1)][rCki+i)\"'"'+'"'

fr(rÄ„_,+l)]' ί;'+ ' 1,2 + ·" + '

Μ''1+ί)]Γ"'·ϊ' [rOi+1)f""' + ""'"2

29

1

pt'C+t)]"1*'+ rl J'2 ++ Γ"ίίη_1

rriri L|\1~1]L' + r~'L2 + ··· + +r_1/-1Γ( ln+l)l1 i

. r«

[rf+Vi)]B'+

[r(r+ 'A|+I)j'' [,r(" +Vι)ίβ,+'"2i

[r(r + +1) ]'*·+ ++

[r(r+1i1+i)]^ [<+,4+Ι)]'"' + 'β»i

Γ r/r + 1 / ι \-ι Κ, + Kl + ... + ΓΚη-ιLM β—ι / J

rr/r + 1 / i \-t K. Li + ... + jtn-1 + LnLM fu+ v\

. <Δ< M"yV f--iV fAV'i;/1' V», ' W "V" "Vy, ' Vy.

, -V. >ί, 'B,+ ·/(, 'K,+'Jf2+...+li

r?)·^ -&£)<%. Vy, ' Vy,

50

2 in 'L, + lL't + .·. + 'Xn-l + *Ln

U'l

+ 'f,

%. +1

+ l^n-1„

/ /+ 'S/, \V, / /+,'2./r Vß. + 'Ba*vA, ,/+v •v%r+i'+"J

C_±_z\Y*. + ri?2 + ... +rÄn-l

^ / + lf°i/r + rL2 + ... + "Xn-l + r£n'\T '+*ϋty* +1

/dnyt + ΛΓ + /^Ι+γ + *B, +' + %\ da" / \ iZr11"1 /

/<*fyr +Λ' + *B, + * + + ...+' +'"Λ <fc2 /

/% + iV + *£, +' + *Z2 + ... + * + XL*. 1 + r +\ dx J ?

tibi, iisdem manentibus aequationibus (2), satisfaciendum eritconditionibus:

!(Jf) ηα+(η—1)δ + ... + 2Α + ί=«? tz=:a + b + ... + k + l,(«Jü) («-Ι)'62+....+2Ά2+'4=«-Ι,

Μ

(1Μα.ι) 2 lkn.t + lln_i = 2 , 1ίη_1 = lkn.i + 1^n-i >

Ι (llH) ηη=ι,j

Ι ('iWi) ηΓαχ + (n-l)'^ +.... +2γΛα +'/χ == w, Γίχ = r«x + rb t+...i • ••+*kl+Tll9

l(rilf2) (w-l)r&2 +.... + 2ΓΑ2 +γ^2 =m-1 7 r£a = γ&24-··.+ γΑ2+γ/2?

If*-.) 2rÄn_i + Γίη—i === 2, r£n_i = Tka_i + r/n_t j'Z„ = 1,

jC+'M,) η'+'α, + («-l)r+'ft, +....+ 2-+Ά, + = η,I '+,i1=r+,«1+'+,fr1 + ... + ' + ,i1

J(r + iM2) (ti-l)r + 1fc8+ ....+2r + lA8 + r+1Z2 = n-l, ' + % =I r + 1ftt + ...+r + lÄe+* + l4,

(r + 1ilfn.i) P + X.t+'+X.t^Q, r + ,U= r + ,*n.I+r + 7n_1,

Facile igitur patet, esse:

/β + 1^ί1 = β^1β+1Λ1,

V+= s^i8 + »fc, "+'#, = (eßi+Bßty+%,(β)

j*+'K, = ·Λ,"+'λ-,, • + tKt = ("B,+·Β,)'+%, . . .\ ... ■ + 'Κ„., = ('Κ,+·ΚΙ+...+ 'Κη.,), + ,ί..,,

52

/s + iLt + ^i? *+iL2 = (sBl + *B2y + %9 . . .

(6) / . · . 8 + ^n-l - (Έ 1 + S/^S + . . . + ·Λ^)· + *1.., ,

[· + «Zn = ('Lt + L, + ... + <Xn_t + *£„)* + Vn.

Praebet autem corollarium 6, si pro u substituitur yr,

2Λ + ι Pro */? aequationem:

- Γ(?ι + 1)Λχ" ~J [/"(« + !)]' + '"'[Γ(»)]Γ+ .. . .{/-(S)f+ '*·[Γ(2)]'+ '

1'

r(r + '«,+i)r(r + ^+i)... .r(r + + 7,+!)'

/d"yr + ixr + *«, /> if-'y, + i\r + 't,dxn ' v dxn~l J

(<?!), +, γ + +,γ + Ί, rf'+\rf.i* J \ Λχ >

Myr +1

cum conditionibus (Γ + 1Ml)j tum, si pro η substituitur η—4,

ajquationem:

dn'lyt= Γ £Wdxnl J j-r^]r + lfc2.... [Γ(5)]Γ + 1/ί2[Γ(2)]Γ + 1 u-

1

r(, + 1fte+i)....r(r + 'A*+1) r(r + %+1) *

55

r +

\ dxn~l ) \ dx1 )/dyr + tV + 1 h d+ *ayr\ dv J ι r + 'fN ' <b/r + 1

cum conditionibus (r + 1ilf2); .... denique, si η =2, aequa-tionem:

Γ ^(5)= r £vJ Γ rtx\f + i*«-idx* J [Γ(5)]Τ + ^0"1 [Γ(2)]Γ + 1/n-4

1

r(< + %_l + i)r(<+Hn_l + i)

+ Λ^ + .Υ + 'ί»-.Y<ry, + A' + '*..t/%r + 1\\ da 8 / \ dx J %+,'+'v'

cum conditionibus (r+'-MTn-i)? postremo, si >1 = 1, aequa-tionem:

<*>/, r r(2)_._^ f-n j

+ ' /n

fyr + l

dx r/o\ir +1 r/r 1

+1 >

[r(2)] r( i„+i)

dy, + t\' + 'h <1 + ';/rfdy, +,V\ dx /

r + 1 fn

cum conditionibus (Γ+4ϋίΙΙ). Quibus valoribus in formulam(«) substitutis, si ad aequationes (2) et (6) animum adverte-

5

54

rimus, inveniemus formulam (4) cum conditionibus (5). Itaque,si verae sunt formulae (l)5 (2) et (5), verae etiam erunt for¬mulae (4), (2) et (5). At vidimus, veras esse formulas (1),(2) et (5), si r = !, r = 2. Verae igitur erunt semper.

Q. E. D.