/dvγ1-1 d l< + lT + '' * + lu

(1)

DE

DERIVATA »;ti ORDINIS

FÜNCTIONIS CUJUSLIBET m

QUANTITATUM

COMMENTATIO

QUAM

EX SPECIALI REGIS GRATIA

ET CONSENSU AMPLISS. FACULT. PHILOS. ÜPSAL.

p. p.

JACOBC8 NICOL. GRANLUND

PHIL. CAN». STIP. KORBERG.

ET

CAROLUS AüßUSTUS NORDBERG

KOBBLAHDI

IPC AUDITORIO GUSTAVIANO DIE III MAJI MDCCCXLVIII.

Η. Ρ. M. S.

P. IL

UPSALIAE WAHLSTRÖM BT C.

(2)

(3)

9

si >o, e termino:

Γ(μ+1) 1

/dvγ1-1 d l< + lT + ^{'' *} + lu

....[r^J^+ ^ +^'-'+i1-1

' ..i.r^iy'Xdx) ^'

dtjxtdziT...dvit-L

terminum; 1ΐΓ(η+i)

Sed nulli alii termini_, qui ea sunt forma, quam in for¬

mula (1) exbibet expressio sub signo J contenta, ubi condi-

tionibus (2) est satisfaciendum^, diiferentiati praebebunt termi-

nos, qui, praeter factorem numericum, erunt aequales ^expres-

sioni (S)9 ubi conditionibus (4) erit satisfaciendum. Subjun-

ctis igitur omnibus terminis, quos jam differentiationibus in-

stitutis invenimus, si ad primam conditionum (4) animum adverterimus, inveniemus expressionem in dextro mern- bro aequationis (3) sub signo J contentam. Itaque, si verae

sunt formulae (1) et (2), derivata (n+l):i ordinis babebit ter¬

minum quemlibet, qui ^ea erit forma, quam in formula (5)

exbibet expressio sub signo tf contenta, ubi conditionibus (4) erit satisfaciendum; nec ullum alium terminum

babere

po- terit, qui, praeter factorem numericum, non ea föret

forma,

quam exbibet expressio (3), ubi

conditionibus (4) erit satis¬

faciendum. Quibus rebus efficitur, si verae sint formulae (1)

et (2), veras etiain fore formulas

(3)

et

(4).

At

vidimus,

veras esse formulas (1) et (2), si η— 1, η =2....

Verae

igitur erunt semper.

Q· E. D.

Corollarium 1. Si μ=f(y, 2,...v,x) est

functio

quae-

libet m quantitatum variabilium y·, z, ... υ7 χj y= φ

(x)i

2

(4)

10

ζ=ip(x)j.,. υ=χ(χ) functiones

quaelibet

quanlitaiis

yarifibp

Iis independentis Aj etx in iiuniero m quantitatum ijyZy...VyX:

. dx d?x d3x dax ^.

ent ^{— =}1, — s=o^- =5 o,... —— —ο, et derivata n:ti

dx 7 dx9 dx3 ($xn

ordinis:

,dauⁱⁱ _ρρ

Γ(η+1)

?

~J

Ϊ

dx11 [r(n+l)]rt+ «+·.·+ «[Γ(η)]&+ß+·!··+ b,

1

+ *-+ .+· + λ +...Η- 1 + Τ

1

I r(a+l)r(a+l)...r(a+l)r(6+l)r(/S+l)...r(b+l).... '

1

(1^ ^' ....r(£+i)r(+i)...Γ(ΐ7+Ι)Γ(/+1)Γ(Λ+^)...r(i+i)r(T+i)

,dav**

,da~y^h ,da'lz,ß sd^vJ

^■dxn/ ^dxn/

^{^dxn/}

^-dx*1'1) ^-dx"1'1

^J ^\Wc

sd*yJ*/d?Zs* rd2f^\1;xdij\bdz\λ

(^v%\ ^::

^dx2^ ^dx' ^dx' ^dx'

^Kdx'

^dx'

■Λ+τ+^...+t

d u

dy dz ^...

dvtdxT

ubi satisfaciendum erit conditionibus:

fn((i⁺οί4"···4"3^+(η~ 1) (b+βΊ*···4·b)-f·.· ·.

(2) ^»...⁺2(A+}i+^...+k)+/+^.««4"1'+ 3Γ5=5wf

Jt—(tbh1_y _t_—■_cc₊β+■^·.+jc+'.λy,,,, t—a+b4*·»·+k+b

(5)

ii

Coroltäriürii Φ Si

é=f(yfifvj

ést fiihctiö qii^libet tri-

11 m quantitatum yariabilium y,z,v, et ij—

<f(x)j

ζ = tp^xjf7

v ^—χ(χ) functiones

quaelibet

quantitatis

variabilis independentis

χ: érit derivata w:ti ordinis:

fdau ρ

Γ(η+1)

dx11

S[T(n+i)]a ⁺

^"

⁺ ^{a[r(w)]6 +}

^ß⁺

^b.r.. ^'

1

*

....[Γ(5)^+"+ ^[Γ(2)Ρ + λ +ΐ

*

1

\ r(a±i) r(«+i) i)itbii). .77 '

g

4

(5)^ ' ....r(A+1)r(+f)r(fc+f)r(Z+i)Γ(λ+ι)r(i+1)

fdny\a/dazV/d"«;

VVd11"1^V/d""1 *y/dn_1i/y

\dxn) \dxv) \dxnJ vlxn-1) ^\dxn-y \dxu~l) ""

f^V\-/'d*z

γ/d2vγ/ιdy\i/dz\λ/dv γ

\dx

) ^\dx2J \dx2J \dxj \dx) \dxj

λ+τ+1

d u

dy

dz?

dv

ubi satisfaciendum erit conditionifrus:

in(a+a

^t^—^a⁺^b

+

⁺

a)-+(n-l)-(&+'/i+b)+...

^...⁺^k⁺^ly ^τ=α⁺^β⁺^..,

+

⁺

2(Ä

^χ⁺

⁺

^{λ] t}

^x + Is)

⁼

^a-fb + i+4

⁺

+

^...

l=»?j

⁺

^b

⁺^l.

Corollarium 5. Si u —f(y, z^ x) est functio quaelibet

trium quantitatum

variabilium

y,z,x, et

y—(p(x)7 ψ(χ)

(6)

functiones quaelibet quantitatis variabilis independentis χ: erit

derivata n:ti ordinis:

dnu_ie η Γ(»+1)

= Γ-

c* J ΓΖΤμ+ΙΥΙ« +"

(s)l

dxm J [Γ(η+1)]α+β[Γ(η)]&+&...[Γ(3)]*+ *[Γ(2)]Ζ + λ + Τ

'

1

Γ(α+1) Γ(«+1) r(b+l) Γ(β+1).... *

1

'

....Γ(Α+1) Γ(χ+1)Γ(1+1)Γ(λ+1) Γ(Τ+1) '

fd^y/d0 z\a/dn-iy\b/dn-i

\<fan) \djcnJ

\<fae"7 [dx11'1/ (dx*J

[dxy [dxj \dx) '

i*+τ+Τ

a u

dydzdxT

(6)

n(a+a)+(n-i) (b+ß)⁺....+^(k+x)+l+X+T=η,

lt^—a+b+..,+Ιί+Ι, ζ—α+β+...+χ-\-λ.

Corollarium 4. Si u~/(y7 ζ) est functio quaelibet dua-

rum quantitatum variabilium y, z, et y = φ(χ)7 ζ = ψ(χ)

functiones quaelibet quantitatis variabilis independentis x; erit

derivata n:ti ordinis:

(7)

15

dnu f* jT(m+1)

dx"

~J

_{[Γ(ιι+1)]α}₊«[Γ(η)]&+ [r(5)]fc +54[Γ(2)]ζ+ λ

'

I

(7).

Γ(α+ί)Γ(α+1)Γ(ύ+1)Γ(β+1)....Γ(Α+1)Γ(«+1)Γ(ί+1)Γ(λ+1)

^dxas ^·άχα^ ^·άχα~1^ ^dx'1' ^dx' ^dx*' ^dx' ^dx-

jt+τ

d u

dydz1

nbi satisfaciendum erit conditionibus:

, \ Κα+α)⁺(«-1)

(6+iS)

+^....+2(A⁺x)+M⁼M?

V8)

(tSS3CL-\-b t » y t=

Corollarium 5. Si u=f(y, χ) est functio quaelibet dua-

rum quantitatum variabilium y: x, et y=sg>(x) functio quae¬

libet quantitatis variabilis independeutis ^x: erit derivata tr.ti

ordinis:

'dDu /** Γ(ιι+1)

ich«

~J

^Γ[Γ(η+1)]« [Γ(η)]*...[Γ(5)]*[Γ(2)]'+Τ

*

1

(9)< ' Γ(α+1) Γ(6+1)....Γ(1ί+Ϊ)Γ(1+1)Γ(Τ+1) '

ίΐΛΊΜ^Υ ^(äVfidjV ^d*+ Tu

\dxnJ \άχη~1}'

" \dx%/

\dxj

dyldxT

'

(8)

(iO) ίΐΛ+(·Λ-~1)ί-f + t~a4-b4-... 4-k +1.

CöröUäriåih1 6. Si ü—f(y) est friiictio qriaelibdf quanti-

tatis yariabilis _y, et y⁼φ(χ) functio quaelibet quantitatis vd-

riabilis ihtfepeiideniis ^x: erit derivata n:ti ordinis:

d"u^u ^Γr Γ(η+i)

?~~J

ΤΪΓΪη

dxn J [Γ(η+ί)Υ[Γ(η)]Κ...[Ρ(^)ψ[Γ('2)γ

(»)

ι

/ ^. ¹

Γ(α+i)T(b+l)

:'./.FKk±l)F(l+iy'

((M\l

^d.Yn'^dxn' ^dxn'^ ^S1Y2J^dx2^ Kdx^\Av)

(ty1'

,Λ,ί '

(12) ηα+(η~^1)^+...+2^+ί==»? t=—a+b+...+k+l.

§. 3.

Ifceb^ma

ΪΪ.

Si ^u⁼f(ij) est functio quielibet quantitatis variabilis y, y=<p{y\) functio fftfajlibet quaVititatis'

vhriabilis

y\, ét y—

functio quaelibet quantitatis variabilis independentis .τ: erit de¬

rivata n:ti ordinis:

(9)

i*

dau v? r(nr( +i) i

.

φχη

J

[r(u+i)]a^r

[r(n)y^ +

1

r(°)J

[r(2)]ai' + &'ia +"'' + h\l +

'

[r(a+1)r(b+1)^.... r(k+i)r(l

+1)'

1 I

[rCat⁺

l)f

[Γ(%+

1)]"

[Γ(% +1)]

1

[Γ('4,

+1)]"

[Γ(% +1)] ^....

[Γ(Ά„_, +1)]

1

[Γ(7,

+1)1"

[Γ(7,+1)Γ....

[Γ(7„.1+1)] [Γ(7, +1)1

du

fd''y\qfd''y\l fd'-'Ί ^λ VA'/ ^Υ

' '

W'·'W* ' ^

fdnyΛ«>ϊ /d*W6' + &'&2

*

\iZa"

)

^\dxn~l ^/

/d*y + &,fc2 + · ·.+

'*

\<ίν

/dyi\a'^« k{ln-l + 1'la

(10)

!(M)

^(lMt) nial+(n-iybi+....+2l1k]L+il1==zn,

na + (η—l)b + .... + 2A ⁺ l—n,

1tir=iiai+ibi+...+lkl+ili9

t=a + b + ...+ k + l,

('iMy (n-l)1^+^....+2^2+ ,/2=W"l,1ti~*bt+... +% +

(Wn.t) a1/^+'U ⁼ 2, ^ ,

\(*M0)

'«.«Ι,

Demonstratio. Corollarium 6 confert aequationem:

dnM ^λ Γ(η+1)

<·"~~J [Γ(η+1)]"[Γ(η)]\... [r(S)f[Γ(2)]! "

1

Γ(α+1)Γ(6+

1)....Γ(Α+1)Γ(ί+Ι)'

^dxn^ ^-dx*'1

^'

^dx2' ^dx'

Ί 1

dy

cum conditionibus (M). Idem corollarium, si pro n substi-

tuitur _y, yx pro y, praebet aequationem:

day ^~ Γ(η⁺1)

dx'~J

_[Γ(η_-f

!)]'"' |T(»)f1.... [Γ(3)]''" [Γ(2)]''·'

1

'

ιγ^+^Γ^+ι)....

ζ-^+ΐΜ'ζ,+ι)'

(dayt\a'f(tfyx\hvdyx\l* d*'y

MW

vdW^

""MW MW *_d %

cum conditionibus (1iMr1)^ tum, si pro η substitiiitur η—1, aequationem:

/dvγ1-1 d l< + lT + '' * + lu

QUANTITATUM

JACOBC8 NICOL. GRANLUND

/dvγ1-1 d l< + lT + '' * + lu

' ..i.r^iy'Xdx) '

babere

forma,

conditionibus (4) erit satis¬

(3)

(4).

vidimus,

Verae

Q· E. D.

functio

(x)i

quaelibet

yarifibp

Γ(η+1)

~J

I r(a+l)r(a+l)...r(a+l)r(6+l)r(/S+l)...r(b+l).... '

(1^ ' ....r(£+i)r(*+i)...Γ(ΐ7+Ι)Γ(/+1)Γ(Λ+^)...r(i+i)r(T+i) *

,da~*y^h ,da'lz,ß sd^vJ*

^■dxn/ ^dxn/

^-dx*1'1) ^-dx"1'1

(^v%\ ^::

^dx2^ ^dx*' ^dx*' ^dx'

^dx'

dvtdxT

é=f(yfifvj

<f(x)j

quaelibet

variabilis independentis

Γ(η+1)

S[T(n+i)]a +

+ a[r(w)]6 +

b.r.. '

*

\ r(a±i) r(«+i) i)itbii). .77 '

(5)^ ' ....r(A+1)r(*+f)r(fc+f)r(Z+i)Γ(λ+ι)r(i+1) *

VVd11"1^V/d""1 *y/dn_1i/y

\dxn) \dxv) \dxnJ vlxn-1) \dxn-y \dxu~l) ""

γ/d2vγ/ιdy\i/dz\λ/dv γ

) \dx2J \dx2J \dxj \dx) \dxj

dz?

in(a+a

+

a)-+(n-l)-(&+'/i+b)+...

+

2(Ä

+

x + Is)

a-fb + i+4

+

l=»?j

b

variabilium

y—(p(x)7 ψ(χ)

(s)l

'

fd^y/d0 z\a/dn-iy\b/dn-i

\<fan) \djcnJ

[dxy [dxj \dx) '

dydzdxT

~J

'

^dxas ^·άχα^ ^·άχα~1^ ^dx*'1' ^dx*' ^dx*' ^dx' ^dx-

dydz1

(6+iS)

~J

*

ίΐΛΊΜ^Υ (äVfidjV d*+ Tu

\dxnJ \άχη~1}'

\dxj

'

?~~J

:'./.FKk±l)F(l+iy'

((M\l

(ty1'

Ifceb^ma

vhriabilis

/dvγ1-1 d l< + lT + ^{'' *} + lu

' ..i.r^iy'Xdx) ^'

(1^ ^' ....r(£+i)r(+i)...Γ(ΐ7+Ι)Γ(/+1)Γ(Λ+^)...r(i+i)r(T+i)

,da~y^h ,da'lz,ß sd^vJ

^dx2^ ^dx' ^dx' ^dx'

S[T(n+i)]a ⁺

⁺ ^{a[r(w)]6 +}

^b.r.. ^'

(5)^ ' ....r(A+1)r(+f)r(fc+f)r(Z+i)Γ(λ+ι)r(i+1)

\dxn) \dxv) \dxnJ vlxn-1) ^\dxn-y \dxu~l) ""

) ^\dx2J \dx2J \dxj \dx) \dxj

⁺

^x + Is)

^a-fb + i+4

^b

^dxas ^·άχα^ ^·άχα~1^ ^dx'1' ^dx' ^dx*' ^dx' ^dx-

ίΐΛΊΜ^Υ ^(äVfidjV ^d*+ Tu

fd''y\qfd''y\l fd'-'Ί ^λ VA'/ ^Υ

na + (η—l)b + .... + 2A ⁺ l—n,

<·"~~J [Γ(η+1)]"[Γ(η)]\... [r(S)f[Γ(2)]! "