ELLIPTICARÜM PRIMI GENERIS

'

·

,

Ϊ£~"

DE

MULTISECTIONE

FÜNCTIONÜM

ELLIPTICARÜM PRIMI GENERIS

DISSERTATIO ACADEMICA.

QUAM

VENIA AMPLISS. FACULT. PHILOS. UPSAL.

MAG. JACOBUS NICOLAUS

GRANLUND

MATHESEOS D0CENS v

STIP. REG. CAROL. JOH.

et

ERLANDUS IIERLÉA

SlDERM. NER.

in auditorio gustav. die v mart. mdccclf.

Π. Λ. M. S.

VIII.

UPSALIAE

excudebant regiae academiae typograpiii.

mdcccli.

1

I 57

>'=14—1 _il'w+1 tlm mm m

S = ,^ΤΤ-2"+

il*i K,»·"-5

II,η

r=I

+(-i}''"1 i»V->n,n~2;'+1 — -f~Cmi

r-n-i ^, nw m ™

g

(~i)rrm

r=i

m »n

-f Ζ,β»·»-5(-1)» -...+(-1>-»L^-l»m"V+I(-I)Ä....+C>m,

ubi Cm, C'm sunt quantilates a numero » independentes.

Determinari autem pqssunt quantitates Cm, C'm ila, ut

r~n-I r-M-t

valöres summarum $ r'w, S (-l)rrw cvanescant,positon=i.

r-?1 r=i

' \

Quum vero in compendiis calculi differentiarum de-

r=n-i

scribi soleant valöres summae S rm posito ro=o, m=l,

r=i r=n-l

m~2, etc., non summae S (-l)r rm, e re duximus propo- r=l

nere aequationes scquentes:

r=n_1

I I

S (-»)··= _—g

(-1)"-

r-l

r=n-i _{2 « .}

r=l 1 4

8

's

(-1)"+jj -1 )*.

ι=Ι

r=μ-i „3 5„

ι·=1

's"

(-1)" + »'ί-1?'

I (-1)"'

r=l

r=»—'I ^„5 tj.,2 I J

S (-i)"*!·5^{= -}

^(-ir+^(-i)»-^~M)" + ^,

r=l

r=n-l 6 r?„Γι μ 3 jt„

S (-irr« ^{= -}

+ U-i)'

r=l

,r=tn-1 ^„ τ 7„β ι«,.»

s

(-l)-r'

-!L ^(-1)»+

(-1)"

r=l

21n2 17 ,N , 17

+ iß

(_1)H +

Γ^Μ-Ι g

s (-1y

r3=---(-l)H + 2rt7(-l)»-7nX-l)»+ i*ri8(-l)H

r=l

^

17n/ ^{^} --γ

(-*)">

r-tl-1 „9 on8 Qlne β«»*

S (_1)1

r5

'-L(-l)»+ ™ ί-1)»+6-^(-1)«

r=1

lS5n' _{ax ,} 51 51

--j-(-»)-+τ(-1}η-τ·

&c.,

39

qaibus inter

computandiim

§ lä.

Formulis igitur? qune

praebent valöres

r=n-l r=n-1

Srw, $(-l)Vro, formulisque recursionum,

in parngrqphis 10

r=l r—i

—15 traditis, adjuti invenimus F(n,42) ⁼-H

+

n® n4 2n2

i?(n, 6,2)⁼

F(n,

4)

+

η8 7/i* η*

JP(n,8,2)⁼

F(«,8,6)

+

+

|7n8 11n* 30n* ₍ 25nJ F(n,8,4)^- ΐ()080 "72ίΓ 1440 840

it*® n* n9 29n*' 2n*

F(n, 10,4) ⁼ F(n,

10,6)

&c.;

n' , 1 1 ^/ 1\-

/(»,2,2) =-γ

+

n4 n®

/(»> ~T2 12'

η4 7»i2 5 η2 5_

/(Μ»*) = ^4—

24+16+

( ) "ΐ«'"4"' ^'

η6 η4 2η2 /(η, 6,2) ^{= -}_jjö

"+*

^ OM n6 llU4 523"2 3 »*, ^s S

/(«A ) "720+288 1440

+52~"06^^'i+

η8 η6 7η4 η2

/(η.8,2)= _{1260 90 180 35'}

w8 11/46 15η4 79η2 μ6 η4 η2

Λ">8,4)= ——-^+^2§8 —

^ö+56Ö(-1)"-^(-1)"+^(-1A

η8 η6 40//4 289η2 η6 8η4 25η2

Άί1' ⁵ ;=5560"120+

960"W2Ö"+48Ö^'1 ^"l92^_1)"+96θ(~1 "J"7

441α Μ8 77/16 45/44 19/42

/(,4,10,4)

=-8100-{

ΠΗ / \ ν /48 7η4 it2

8040^"^ "^560^"^ 720^"^ "^440^"^"'

λ μ λ fi\

29rtl°"

83"8

+

+

η8 η6 7/44 5η2

"

2688

^

^1^"'

&c.$

η2

~1Γ

1

+_4~

ί44 7η2 5 >42

i- χ

'

24 24 ¹ 16 8 η4 η2

12~

"

12' ^..

η6 11η4 525/ι2 , 15/4

, . «G _, W4 15η2 η4 η3

9 '

120 -16 240"48 C~1>V

61

n6 2^il2

φ(η,β,4)^{=■- —}—»

φ(η,8}2)=

η9 η6 49η4 289η2 η6 5η4 25η2 '5360 120+960

6720+480^~ 1^ί»·2(""1)η+

tr η2

fl fl fl^ 11.4 yj^

η8 η6 7n4 η2

φ(η,8,6)= _1260—

öö+

29n10 85n8 55η6 4661η4 453n2 f(n,10,4)=-5024oö

+4Ö52Ö

560Ö"^~

η

Ϊ55<-1)"+Ϊ92(-Ι)η

2088^

7n4 584

5η2

H)"+224(-l)·.

,10 77η6 45η4 19η2

φ(η,10,6)=

-8|00+452

+

,6 rr»,4 .,2

ησ nu 7 ιγ η"

C-OH^c-ir-^önC-DHnöC-1^

5040^ 1 360ν ~aj 720

&C.;

^(η,1,0)=η,

ψ(»>3»0)==ψ(η,5,9ί)=

"'"ö"+J2+4+1)^

V(n,5,0)=V>>s»4)⁼

η³ 73η η3 ' 11η,

-ίτιΗ)η+—(-*)%

120 8 240 24 48

ψ{η,Ζ,2)=

η& η3 51η η3 5η

1 (_ I )ηΛ (-1')η 50 6 1 120 12 7 *24

η'

^7,0)^^,7,6)^^ +

13η5 149η3

( 281η

3040 ' 1440 7440 ₁₁₂₀

I η<> ^{t ·\η} 55η +

^(-1)Τδο(-1)"+ΐ^(-,)"'

η7 η5 67η3 157η

ψ'η.7,2)=φ(η,7,4)⁼ {— [--——

^ ^ 5 '

560 1 52 480 1 672

η3 5η? 47η

4 £-1}η ■(-*)""i—'— (-Ι)ηι

06 52 ^ 96

η9 2η7 49η9 22124η3 5505«

t//n,9,2)η '⁼ιί/(«,9}6)ΎΚ 22680 945 [-1728 . 181440 11 20160—

η7 41η5 55η3 1051η

' 2016

^ +2880ί~,')

( ^ +

(~J>

^ η9

25η7

ψ(η,9,4)_{=-60480 10080 960 ~~ 15120 4480}

—7 17η5 25η3 669η

(-*Τ+ (-*)η-τγλτ: -4)η+ —(~4Λ

4120 960 ν 240 λ / ' 4480 247η11 73na 421η7 2551η*

ψ(η, 11,4)^{Vv ;=}φ η,11,6)^{= +}

19958400 725760 201600 90720 597301η* 25265η η9 165η7

+ + — (-1)'

5628800 177408 145152ν J 120960 ;

17η5 66407η3 2125η

~\ (-Iiπ /-!)"+ ^- -1Λ

864V ' 725760 ; 16128' '

<5cc.;

sive, ut aliis formis utamur, l:o si η est numerus: rinpar,

ψ(η,Ι,β) ⁼

*

=—y

*

)

ψ(η,5,0)

1-2545 ' v / 1-2-5-4-5-6-7

2:o si η est mrmerus par,

ψ(η,1,β)= ί,^(»;5,0ϊ= ψ(«,8,0)

65

_»(„"-«·)(»»-* O

ψ(ί1,7,0)

I·2·5·-4 5 1·2·5·4·3·6·7'

unde prodire videntur formulae generaliores

v ' ' w

1·2·5'4·5·6·7....(2rJ ^ si η est numerus impar;

V' 1 ·2·5·4·5·6·7 •••(2r+l)

si η est numerus par;

quae formulae evolutionibus cognilis functionis sinnu se- cundum dignitates adscendentes quantitatis sinu conveniunt

et convenire debent.

- . '\

Eodem modo efiieitur, ^esse,

l:o si η sit numerus par,

2 ^ r»1\

r λ η , . η in -2-

,(»,2,ο) = - _, ίΟα,ο)=

12 5 4,

,(η)6ιΟ)y = -

1·2·3·4·5·β

2:ο si η sit numenus impar,

/ „ »»2-ia ^{/ α η\} (n2-l2f(«2-52) -

,(„,2,0) ^—— -jT^ '

l(nJ<°)

φ,6,0) ^-- _,.2.5

·4·5·6 ·

C''

unde prodire videnlur formulae generaliores

r,r «V-^XnM^u'-e*) 4»Μ8τ-2)']

,(n,2r,0)=(-l)

(2r)

si η est numerus par;

τ(..^y ...λ)^' -,

(»'-Ι'.(»'-5·)(»»-8»)..·[»'-(8Γ-υ»1

si η est numerus irnpar;

quae formulae evolulionibus cognitis funclionis cosnu se- cundum diguitates adseendentes quantitatis sinuconveniunt

et convenire debent.

Praeterea facile sequStur, esse

F(n,å,2) =-2. ,F(n,6,2)=2

anVa

*a

/ v ; 1*2- 5*4 V 7 1·2·5·4·5·6

*M,2); = - &

1·2-5-4·5-6·7*8 '

unde prodire videtur formula generalior

Fin,2r.2)^ ' ^—(-1Γ1.22r'8

n^n "*0(n2^2)(w2"oa)"-[ft2-(,'"^)a]

t-2-5-4-5-6-7-8 ^... (2r) /(u,4,2)=2 , /(n,6,2)⁼-2

,Vy 125 4 y ^ ; 1-2-5 4-5-6

/(„,8,2) «

2^>2-i2K^(^2)>

•7 v y

1-2-5-4-5-6-7-8 unde prodire videtur formula generalior

/Yn,2r,2)= .

1·2·3·4·8·6·7·8.... (9r)

'

l:o si η sit numerus par,

qt(n,4,2)⁷=2.1.

η^·η~^}

12 5-4-56

*(n,8,2)^' = 4.5

*V-*>'-flaKna-4')

5-6 7-8 '

2:o si ^u sit numerus impar,