SUMMATIONE SERIERUM, m

m ^{D E}

SUMMATIONE SERIERUM,

S E C U N D U M

DATAM LEGEM DIFFERENTLVTARUM.

C O N S E N S U

AMPLISSIMI PHILOSOPHORUM ORDINIS

I N 0 N I V E R S I T A T E C A E S A R E A L I T E R A R U M D 0 R P A T E N S I ,

MODERANTE

D. JOHANNE GODOFREDO HUTH,

M AT H E S E O S P U R A E E T A P P L I C A T A E P R O F E S S O R E O R B I N A RI O , HOC TEPPORE DECANO,

A U C T O R

C A R L H E I N R I C H K U P F E R ,

M 1 T A V I E N S I S.

MITAVIAE ,

I rn p r i m a t u r,

H ^{u t h ,}

p. t. Dec.

0 >^f

D o c t r i n a serierum campum late patentem praebet, magnam copiam meditationum evol- vere; atqiie cum tanti momenti sit in Analysi, contemplatio serierum tantum utilitatis quan¬

tum voluptatis adf'ert. Quare meas meditationes huc spectantes, qiiamvis exigiiae sint, in hac Dissertatiuncula exponere constitiii: ac ne hujus libelli fines excedam, ad eas series me restringam, quae secundum quandam legem differentiatae e v a d u n t Sit scihcet propo¬

sita series.

4. f x

4« etc.

a 4> b x 4. c x* 4« d x

4« e x*

cujus summa = ^>x habeatur

quaemultiplicataperx

'etdifferentiata, p e r q u e d x d i v i s a , dahit a n ' x " ' - 4. b(n'4^) x

4. c(n'4<3) x

'

* 4» d ( n ' 4 ^ ) x

quae multiplicata per x

" ac den110 diff'erentiata, dabit

* e(n'4^)x"'+'

^

, i

n " )

» ' - ' + ° " - ' 4« b ( n ' + i ) (n'+n") x"'+"""'4<c(n'4.2) (n'4<n"4>1) x"'+""4-etc.

quae rursus multiplicata per x

" ac differenriata, et sic contiiiuo multiplicata per x

x

n

. . . anteqiiam differentiatio s11scipiatur, perducet ad seriem hrijiis formae:

a,n'. (n—i+n'Q

(n<

—i4*n"—i+n'"—i+n

) . . . .

4. etc.. ..4«p( 14-n'4.m) (n'4m"+m) (n'— i +n"+n'"4*tn).... Q£(n— 1 )4^4<m)

i(»-'H°H-i

. . .

t

. Q u u m f a c t o r e s , quibusCoefficientescomponuntur, i t a s i n t c o m p a r a t i , u t , pluribusn', n'"... positis = 1 , aequales e v a d a n t , haec forma seriei prodibit:

y J ff* ß y j&

a m . m . - m , . m , . . . m , ' x 5 > - " + b(m+i) (m,4<i) (m

4u) . . . . ( n v f r i ) x^<"-'>+

4. etc.

n",

x ß 5

m .m. • m,

4.p(m + k)* (m,'4-k)

... (m

4«kf' x^"-

)+* +

;

existentes«, ß, y . . . m; m , ; m

; . . . numeripostitivi etintegri.

Tradidit illitstris Euler in ejus Institiitione Calculi Differentialis methodos, quarum ope series et hac forma contentae siirnmari possint, d u m m o d o # , ß, y . . . sintnumeri determi- nati, plerumque tarnen illae, ab evolutione differentiarum finitarum vel differentialium pen- dentes, gravioribus difficultatibus obvolutae sunt, dum seriec generalioris indolis sint, earumque summae ita assignandae, ut ad nulIas ampIius evolutiones neque difFerentiarum, neque differentialium perducant. Quare cum ad methodum, nondrim in summatione serie- rum, q u a n t u m e q i i i d e m s c i a m , admbitam, pervenerim, c u j u s o p e f a c i l i i i s s u m m a e i s t a r u m serierum generahoris indolis eruuntrir, operis pretium i"ore videtur, eam i n hoc tibeüo expo-

etc.

(R)

1

n e r e . P r i m o

q u i d e m , ut sponte appareat, quaenam

in usum vocanda sint,

accuratius perpendam seriem:

i 4 . 2" x 4 . 3n x2 + 4 " x14 . 5" x * 4 . 6" x ' . . . + f x ' - ' flfc«) c u j u s

summa

multiplicata constat esse =

x ' ^ — t » - x ^->+*'*x n ( n - Qt0_ , _ x > *4x ' + x n ( n - i ) ( n ^ ^ . , ^ ^ c

V x — i ( x - i ) " 2 ( x - i ) * a. 3 J—

quae reddat

0, si ponatur t =

H a e c

expressio reductionem

quam quidem formam

e seriei con- templatione delineabo. E t

summam assignabo, si ponatur

et series

infinitum excurrat, ac deinde si pro liibitu abrumpatur, et

induat valorem.

Q u a r e

duae

investigationis

erunt.

§• 2.

fnvestigatio nova seriei

SL ponatur x 1 et series in infinitum excurrat.

1. Q u o casu constat esse

1 1 4« x 4« x² 4« x³ 4« x * . . . . 4< etc.

(R')

1 — X

( I ^- * ) '

— = 1 4« 2 x 4 . 3 x² 4« 4 x³ 4* 5 ^x* 4«

. . e t c . . . .

- = H . ( i * Q ) x + ( 1 + Q + 3 ) x² + ( i + 2 + 3 * 4 ) x ' * 0 + 2 + 3 + 4 + 5 ) x * . . . . • •

(*„—*)'

q u a e

series

(1

(1—x) * 4 4 1 + ( 1 + 2 ) + O + 2 + 3 ) + ( 1 + 2 4 - 3 + 4 )

= 1 4. 2* x + 32 x2 + 42 x3 + 52

x* . . . etc. . . .

q u i a c o e f f i c i e n t i p r a e c e d e n t i c o n t i n u o

additur

etc. qui componunt nu¬

meros

qiiadrati

evadunt.

2 . J a m i n d e s u p p o n i p o t e r i t , seriei

j 4 . 2³ x 4 . 3^$ x² 4« 4³ x³ +

'/ x* . . + etc. . . .

ac

m o d o

(1

« V> tL* V1 4 . 2* x 4 . 3⁴ 4 x2 4 . 44

x

+

etc.

fx

F x ( i — x ) '

sicque

ita ut

seriei

sub forma

^ x ,

fx functionesipsiusx; siseriesprolubitoabrumpatur, istisfunctionibus

erit,

q u o d sit = X. Posito x =

potestatum n u m e r o r u m abtinsbitur

3

Quodsi vero ponatur x = i erit ( l — x ) = o , i n d e , sumto f i n i t o t e r m i n o r u m n u m e r o ,

c ^ x

fx + X o

fien oportet =rs - , cujus ergo valor differentiatione usque ad ordinem (n4<i) in-

Ci—x)

o

stiruta, e t t u m p o n e n d o x = i e r u e t u r : differentiationeveroproducta, ex(n4o)factori-

bus composita, evadent, quem niimeriim factorum e t f u n c t i o , q u a 2 x " e x p r i m i t u r , conti- net. Quare c o n v e n i t c o n c l u d e r e , denominatoremhabituriimesseExponentem(n4<i).

3. Si fx definito terminorum numero coustare accipiatur, e serie (R

) multiplicata per ( i — x )

expressionemfinitarn prodire o p o r t e t , quametsi series in infinitum excurrat, vel, quod idem est, coefficientes, datum terminurn excedentes, in producto evanesceredebent.

Instituatur Multiplicatio:

i 4»

^x vf 3

x

* ⁴

^x

4- ⁵

^x* 4< ^6"x*

—

(n+i) — (n+i)i

—(n+i)3

— (n+i)4*

(n>fi)n (n*Qn. ^ (n*i)n

(n*Oa*

(n+i)n

4. i C-4n

* (n*Qn.

2

(n+1)n(n-1)

— 4 2- 3

^ (n4*i)n(n-Q(n-2)„

2- 3- 4 ,(n4-Qn(n-i)(n-aX"-3),,»

2- 3- 4- 5

(n+ ])n(n-1)(n-2)(n-3)(n-4)

4< ^7"x' 4. ^etc.

(n+i)6"

2- 3- 4- 5 . . 4. pV'

-(n+i)(p-i)"

.(n4*i)n, h

(p-2)"

*

(n+i)n(n-i)

2. 3 (n4u)n(n-1)(n-2)

<P-3)"

(p-4)*

2. 3- 4

* fc-(n*0)»

2 .

3. 4. ^S 6

Q u o d productum littera Q designaho.

Ejusmodi Coefficientes, quibus quantitates x

, x"

et sequentes affectae, utique eva- nescere, j a m e doctrina serierum recurrentium apparet, quod attamen nova demonstratione illuStrabo.

Sit p

Coefficiens quicunque seriei (R

) inde a termino ( n + 2), erit, existente k > n, Coet- ficiens ipsius x

in producto Q hujus formae:

P- — (n*OG>-0" * Cn4.i)n

_

_

q»ü, evolutis Q? — i ) " j (P — 2)

/

2 2. 3

; G>-3)";

(n+i)n(n^O^ 4. (p-(n4.!))»

etc. formam sequentem induet:

2

etc. etc. etc. etc. etc.

( n * i ) ••• * r p - - - ( n * i ) n p " - ^ ( n A i V ^ ^ p " - ' - r n 4 . i l n r n - _ i ^ r n - Q y -3 * + ( n + 0 " p " " l

a . ^ . . . f n + 01- , 2 . 3 J

2 . 3 . . . ( n + 1 ) L 2 ²- 3

C 0 n s i d e r e n t u r n u n c j u m C 0 e f f 1 c i e n t e s , q u i b u s q u a n t i t a t e s p " , p "^-' , e t c . a f f e c t a e . ( n 4* 1

C0eff1ciens ipsius p

e s t = i — (n 4 . 1 ) 4 . ^ i 4« 1 = (1 — i)"+* = 0 . Coefficiens

1)"=0

2 —

sit m o d o n > o.

HucusqueergoCoefiicientesevanescere patet; erit demonstrandum,

in

esse s=3 0 , si in genere

, n ( n — 1 ) ( n — 2 )

2 . 3 . 4 * • • • — ⁰

denotante n ' n u m e r u m quemcunque positivum, integrum nec

n.

„/ n . ./ n ( n —

1)

1 _ ¾ _ 4 . 3 * \ ^J

2 2 . 3

4"

Quod manifestum» existente n'=s 1, quare videndum erit quomodo aequationes ^sequen-

tes a praecedentibus pendeant. Accipiatur ergo aeqiiatio

„< n

^. n(n-i) , n(n — i ) ( n - o ) . . ( n + l ) " '

i o"> _ 4* a"> _i 1 .n', — c . . . . M i—*—<_ ss O

2 a. 3 4

3. 4 - n+i v. 3.

quae locum haheat, dummodo ne sit n', > n

quae, multiplicata per (n ^4« i), simili modo erit = ^{0. Quare} multiplicetur per (n4*i), vel,

quod idem est, termin11s primus per ⁽¹ 4* n), ^secundus per ⁽² 4<(n — 1)), tertiuc ptr (3 4. (n — a)), ^quartus per (4 4» (n — 3)) et sic porro; obtinebitur

1 — a"''+

" *

~ — 4"'-+

"(

- 0(n=ja3 . . . , +• ^^'•+'

* 2 2, 3 * ' 2. 3. 4 ^{- n+1} !

* Tn-Q"'- "fo"

4.

"'.

C

-OC"-2) , n(n-Q(n-2) (n-

) n<n

L ² 2. 3 2. 3. 4 - n ' J J

Cum ^pars secunda, sub parenthesi conteiita, sit =

„n _

.\ ^Z_i 4. 3-'. C"-OC^2) _

<(n-i)(n-2)(n-,) _ T _

L 2 2. 3 2. 3. 4 J

propterea qirod evanescit, si ^demum (1 — i)

'' = 0, inde sequitur, et primam partem j

»',+i £

"(" ~ O A»'.

-» n(n — OTn—•Q .etc. esse == Q /j\

2 2. 3 2. 3. 4

'

Quo modo ^vero pendeat

ab aequatione (1 — 0"

"' ^{= 0} sequenti modo clarius apparebit:

Designent Ci — Of; (1 — Of; (1 — 0,"; (

— i)"quantitates i — 2

- 4- 3*

(

—Q _

= m(m-i) (m —

)

2 " " 2. 3 ^2. 3- 4 ^etc.; 1' . j m . . mfm— 1)

2

V *

2.3 *

-;

. m . . m

— 1) 1 ^{^ _}

m . „ m(m — 1)

i-2*-*3* \ -^ + etc-; 1 2" - * 3 0 . - — etc. • • •

2

3 2

Quibussignisadhibitis, ratio, quaminter se quantitatesconstituant, Sequentitabulare- praesentari poterit

(»-On-

(i-On-,; o — os=:-

O

OS-*; (' — OnZl; O - i »

(1-O"-3; 0-08=1? (1-0¾ 0 — 023-

O -1^_.; (i -OS=:; O -1/-_:; Ci-OS=:; O — OS=t '

O — (i — OS=Ji O — OS=,; O — OS3; (i etc. etc.

OS=;; a - OS=i (i-0"; 0-Or'; 0—0;

C1—1)";..C —>>"7*5 ('—O-

etc. etc.

0—or

; (i—or

; o—or

.-- a —o:-

v*-v 1 v—;. > v-'>» * 0-0""'> (i-0""

; (i-0"

; O—0"~* 0—0

Quilibetterminus superior(i— Ol

duobusterrninis inferioribus (i — 0*_, e t ( i

1)*3J

pendet, accum quivisserieiiiuimaeterminusevanescat, omnesusque 2 adsupremum termin

. o p o r t e t f i a n t

= <s. Ulterius

licet,

quod ( l

aequa*

t i o n e ( 1 — 1)° = i

demum

E x m o d o

allatis

C0eff1cientes ipsarum

p"-

, p

q11are

quemcunque C0eff1cientem

4 .

Quodsi

Coeffi- cientes producti

induere,

S i f i n g a t u r s e r i e m i n i n f i n i t u m

excurrere, C0eff1cientes

producti

infinitum

e x c r e s c e n t , i t a u t f i a n t n u m e r i

inf1niti0rdinis

*|

sin| cjignitate infinita

q quemcunque

+ q

positivum,

minimum,

A x

_ J

X '

s t a t u i m u s

^ ¹ *^ X*y>q+*&-*kq'

derKotante A numerum finitum;

quitefmmus",'denominatore e x i s t e n t e i n f 1 n i t e m a g n 0 r e s p e c t u n u m e r a t o r i s , u t i q u e e v a n e s c i t .

? 5. E x a l l a t i s c ö n s e q u i t u r , f x

definitoterminorum,

numero

f x = 1 4 . A

x 4 . A

x* 4< A, x*4« A

x

. . . . 4 . A

_ , x"

'.

q u i v a l o r e s A1, As, A , , e t c » . . . f a c i l l i m e e x p r o d u c t o Q i n v e n i u n t u r .

P e r s p i c u u m , e s s e A n . , = = i , A , ^ = r A , , A , , . , = A a , A , , . , = A j , e t c . ;

Coefficientemenimtermini

(n4.2)inProductoQevanescereostendi, quare, (P — P)">

(P—(P*—0)%

X X JL" '

f-P 5^^

^{^}

- antepositis

QV) ^ ^a

hanc formam induat

ac postea

multiplicatione,

Cpeffi- ciens = 0 , atque simili

Coefficientem

Hh i)'in

termini

ter-

(y\ + j 1 % Yi

m i n i ( n — i ) = o 4 > 2 "

—

—

6. E x

allatis nunc coUigitur,

(R')

— i 4 « A . x 4 « A,x* 4 » A , x * . . . . Aⁿ. , x " ~ ' e x i s t e n t e X < 1 e t s e r i e i n i n f i n i t u m e x c u r r e n t e . (1 X ) .n+i

P a r s s e c u n d a .

Si series pro lubitu abramparur, et x quemcunque induat valorem.

DecignetXfunctionem, functionifx addendam, quoserieisummaabruptaeobtinea-

tur; sit t

x *

terminus seriei ultimus, et consideretur pars ultima Producti Q , quae erit:

f¹. x^t_I

— ( n + i ) ( t - i )n

» ^ ' > ( t - Q '

— (n^i)t"x'

^ C 2 t _ 0 " ( t _ 0 "

2

* C " * Q " t " x ^

( 4 ' > ( n - 0 ^ ^ . ( n 4 . i > M , ^ 0 " ^ 0 ( t i Y C ^ O < " - O f y ^ * etc, r\ ^ ' n ' s rt

«• 3 etc.

«• 3 etc

2- 3 etc.

2- 3 etc.

I n d e

sequitur

( t ^ i ) " x * 4 - [ ( n 4 . i ) ( t + i ) " - ( t 4 > 2 ) " ] x « +¹ ^ ^ ^ ( t 4 > i )ⁿ~ ( n 4 > 0 ( t * 2 )ⁿ4 . ( t 4 . 3ⁿ) j x * +²

^

r(nfri)n(n-Q^

L 2 . 3 2

J

- r f o * P n ( n j ^ X n - O f t g^{i ) n} _ ( n j , , ) n ( n - i ) ^ ^ ⁺

&**>cy*tf

L 2 . 3 . 4 2 . 3 2

— (n * i ) ( t >p 4 ) - * ( t * 5)"]x'+

. . etc.

Q u a m f u n c t i o n e m X e p r o d u c t o q u i d e m v a l o r e s t " , ( t — i ) " , ( t — 2 ) "

etc. continerein-

v e n i t u r , s e d

facile

ejusmodi, ad primos terminos ipsius

magis accommo-

d a t a m^>f o r r n a m ,

sicuteamexpressi,

§• 3 -

d .

x

'. fox „_

Z_ n ' x " ' ^ x Hh x" _

d x ^T ' d x

d.x"". d. x"'.ftx_^vn,_,+n„ e"^"^^''x^^

r * » J , , 1 * j . .a

d x » ' • ' d x " * d x^s

d. X"'". d.X"". d.x"'. ff* ^e///xn-_^I+n''_^I+„»'_¹⁽^^x » p <//gn'-i+n"-.+n"'d- @*J^ ///^n'-i+n"4^"'j'^

CÜ* ^ * ² dx^a

>ft e/"x"'- «_a_„/» d M £ x

dx1

etc.

Designet

etc.

(

n m J e j u s m o d i

differentiale ordinis

T j )

sr= e* x * @x Hh e f

* d

d x ' ~ ^y~ ⁴ <*x d x '

etc * e|x'H-P (d d.xHh d^sx* Hh d^m x^m)^% d*+' r

r I 1

d . @x

— - T l — eP H - ' . xf f\ £ x Hh e ? ^ - ° xr f ,H - ' .

d x »- 1 -' d x

>h etc. . . . ^ e ? ^^-0 xf f , H _ p. ( f ' x )^V

»j<(f*)V-.

Designet in genere

fiinctionem ,

terminorum numero

f u n c t i o n e m q u a m c u n q u e

ipsius

Manifestum est, functionem fx

si

s e r i e i

propositae T ad formam —

U

Ql

cognitam ipsiiis x ,

<&x

q u e a t ; p r o d i b i t

enim aequatio fx ( T

$ x , Jam

haec forma

tur, investigabo.

Res eo redit, ut inveniatur

modi

quae

l u t i

inde a q u o d a m termino assignabili reddat = o. Sit ^ x ita coinparata, ut

t i a l e

ordinis p determinati sit =

Y

(d Hh

d,

x

d ^{t x}

^x

^{; =}

^ x differentiationeeva- dant series formae (R) (§.

denotante v

positivum,

vel

Instituatur

legem ( § ,

i n d i c a t a m , ,et

d , +

"f n^)

V x

J ^{d. @x}

- — f?H-»)

*a tf5x i p(pH-=) /

^iH-a —

^{. * e,}

* e<^+<

-»H-P (f

)V

r»)

«r

+pH^

_

j^,_+_pH_^ (f/^V-!

* (d<,">

X"+P-M * dJ0

«+PH->

^{. , .}

d£2-, 1

ds

"C^0

dx

^-* e

^

. . . etc.

etc. etc.

dx

d. ®x

e

. x*"-* ^x 4. ^e? x^"-J^-. —J- ^{etc. . .} 4« e' x"-»'.+' (f'x)

• • 4« d<"-P) ^{x^"}

^{*^*^"}

^{) ( f x )}

4« (d<

°

* "

- ' 4« etc.

4. (df"-*>i x'"-*-He+*4, • • , • ) (f'x)

-

4< etc. . . .

4- (d^x^"-pH-P-h-(n-p) ^

_ ^ ^j^<rn-p_f-pHKn-p)m^ ^^V-(n-j)

quod sit = Qf 4. F (fx)

4« F (f'x)

-

+ F" (f'x)

~

. . . . F"~* (f'x)

^

-*>

quae forma ad hanc reducitur

<H4<(F. (Px)—* 4- F' ( f ' x ) " - ^

4> F" (f'x)

-*^

. 4, e t c . . . . * F—'-

. f x ^ F - ' ) quae evoluta hanc debit formam generalem.^

C ^n-PH-P Q

^

-P-I-, Cx

4« etc.

U +

^a_p_V

* C^{( n}_ j^{) m}x^{t f n}- ^ ^ " - p ) «

quae sit fx

(fx/-"-

Litterae e, d, a, C scilicet cum e o r u m signis denotant quantitates, perdifFerentiatio- n e m et multiplicationem introductas.

Si nunc sit proposita series ejusmodi formae, ut (R) ejus summa sequenti modo assignari poterit:

fv

E dijferentione ipsius ^x deducaturforma surnmae, quae sit — 4* 9(} ac deinde po-

<fcx

natur fx = ((R)-U) $x j (0

3

Comparentur singuli termini, cumque ita pro quocunque Coefficiente nisi unicus detur valor, et x in ipsa fx potestatem a n — p 4 ¹ P 4" (n — p) ni «o« excedat, omnes Coefficien-

tes Producti ^((R)—Qf) 4>x w//ra terminum, potestate crn p 4 p 4 * ( n — p ) m ipsius ^x Jectum, progredientes neussario evanescere debent.

Quod attinetad functionem 5(, ea sine difficultateinveniri poterit, n a m , cum p sit nu­

merus determinatus, differentiaha^P: ^_2: . . . assignari parsunt; adquantitates

dx dx»' d x

dx?

e v e r o , quibussuntaffecta, abtinendas, consideretur, eosdemCoefficientesevadere, siloco

^)x differentietur e

simili modo. Designet E functionem quantitates e involventem, d"^

n^

^e

\

seriem, q u a e~

exprimitur, itadifferentiatarn, prodibitE = ^XfXV

"' Q

aequatioda- bit valores e.

Jam nunc qnantitates C; C,; C,; C , ; . . etc. ex aequatione (i) sine difficiiltate definiri possunt, si m o d o $ x ita comparata, ut ejus evolutio generalis vires Analyseos n o n superet.

E x e m p 1 u m.

P r o p o s i t a s i t s ^{e r} i e s c o g n i t ^{a .}

1

^l- 5 £ ^{* etc.}

Arc Sin y = y 4» i +

3 a.4 5 2.4.6 7

e qua assignetur summa $ seriei

* L 4. _1_ + yJ? 4, ±rr^-S ^a - j* * T 3, 5. y». 7_' + etC.

Y' 2. 3

4 5 2. 4. G 7

4. 6. 8 9

Differentietur — Arc y, q u o facto obtinebitur

y

— — Arc y 4. L _ _

Y y - ( i - y ' ) S

E r i t 9 ( = 4<2"-Arcy, — Y" 4. sinpar, — siimpar. p = i ; o-n — p = —3, quare c n — p 4. p = 3 — 2; m = 2; c n — p + p + (n — p) m = 2 (n — 2); v = — | ; erit igitur forma

C

A * C, + c

y

...*C„_

y

-

Multiplicetur per x et denuo differentietur, sicque porro.

summae s = * 2" — Arc y 4« __^

— y'

1 + 2" 1. 3 (3

^4< 2»)

«• 3 2. 4. 5

Cotfficientes C, C, etc. aequatio.

0-y

) 2(n—1)4. 1 _ Ponatur 1 4 2" = a;

2

2

(n — 1) ^{+ 1}

c etc. : = fj,, prodibit ad determinandos 2

y" c,

- + b + Y"

cy

*

a

+ yua 4> jub tf

4. Afci2a 4.

Q Y

dy'

C

Y

ey'

1 2

*c

-,y*

-"=:

1

2 2

§• 4-

Saepenumero series occurrunt, de quibus n o n statim perspicitur, quomodo earum sum- mae ad formam (§. •>.) exhibitam reduci queant. In usum t u m erunt artif1ciav0canda, quo¬

rum ope aequationes obtineantrir, quae ita comparatae sint, u t quantitates determinandas in altera parte signi aequalitatis separatim contineant. Quod quidem in genere interdum com- paratione plurium serierum, interdum introducendo novas series, quarum summae quantita- tis quaesitas continent, efficitur. Quem in finem consideretiir series:

. 4« <JL x' 4« . . in infinitum 1 — i - x

*

- — x

Ü _ 2. 3 2. 3. 4. 5

3. 4. 5. 6. 7

functio x Cosx differentiatadabit

2. 3. 4. 5 . 6 . 7. 8- 9

Cosx — x S i n x s=s 1 etc.

J l x

* . 1 ^x'

2 - 3 2. 3. 4. 5

qiiae multiplicata per x, et differentiata, perque dx divisa dabit

<J*

Cosx 3 x S i n x x*Cosx p 1 — ... .. x

4*

b ³

2- 3

2. 3. 4. 5

x* — etc.

eritque in genere (1 4» Cx' 4. C,x* 4« C

x

= 1 — ^L_ x» 4.

2. 3 2. 3. 4. 5 Erit que simüi modo

C

X ^

) C o s x 4- ( C x + C J x

* Q , x

~

) S i n x

X*

— '- x' 4 . etc.

2. 3. 4. 5. 6. 7 ( 0

Sinx + xCosx r= sx — AL- 6

2 .

3

3. 4. 5

Sinx 4» 3

Cosx. — x* Sinx = 2*x

etc. . 6

JL_ x' 4.

2 -3 2 .3.4.5

etc.

sicque porro

( i 4. Cx

4. C,x

. . . . + C

x

- ) Sinx — (Cx4. Q x *

• •

4« C

x"-

) Cosx =

*^B

2»- X _ i_ 7» +

ßn-l

3 etc

2.

3. 4

denotante scilicet n numerum positivum ^integrum imparem. ^FaciIe perspicitur ^quomodo se habeat, si n par.

Aequationes

per Cosx, (2) per Sinx multipHcatae et additae, ^dabunt 1 4. Cx

4. C, x

. . . . + Q x*-', (quod sit T)

1. 2 .

3. 4. X

6

1. 2.

3. 4.

_ f

1 . 2 .

3

r.—I

1 . 2 .

3

x

4*

Cf.

1 . 2 . 1 . 2 .

3. 4

1

1.

3. 4. 5. 6

6*-'

1.2.

3.4. 5

1 . 2 .

3.

3

Qn—i

1. 2.

3. 4. 5

etc.

4« etc 4« etc 4* etc etc ....

4

etc 4< ^etc

Inde quo lex, ^secundum quam CoefficientesC, C, etc. progrediunt11r, facile perspicitur.

Ponatur n 1 = n', eritque

c = _ t±L * .*

c.= ^*^-

1. 2.

3. 4

_ 7 " ' * »

~ ~ . - - . . C .

"'Cf .2»'

1 . 2 .

3 5"'*3"'

1, 2.

3. 4. 5. 6

3. 4.

etc. etc.

6"' 4»

1. 2 .

3. 4. 5 2. 3.

3

x — x

+

1 . 2

- 5 "

1

^J- ^{* *}

1 . 2 .

3 . 4 3

' 1. 2. 2. 3

1 i-a. 3- 4-5

- x ' 4< etc.

i,

3. 4. 5.6

— 2'

' X * i ^ - X

— 2- 3

6°' 2

1 . 2

2 .

3. 4. 5

^n'

2.

3.

.x' 4«

1. 2. 3. 4. 2. 3 3*

1.2. 2. 3. 4 . 5 1

1. 2. 3, 4. 5. 6. 7 8'"

CE

2.

3. 4. 5. 6. 7 6"

2 .

3.4. 5.

x

— etc.

etc.

quare

C

= 1 — s-'

V' 1

Q = — A _ —

4 . 1. 2 2. 3

Q = *

c ; = -

'

1 . 2 . 3 1.2

3"' 1 6»' 1. 2. 3. 4

7* _

1.

3. 4. 5. 6

1. 2. 2. 3 2. 3. 4. 5 etc

1 . 2 . 3 . 4 . 5 1 . 2 . 3 . 1 . 2 1 . 2 . 3 . 4

r _ ^ = L

Li.2.v.(n-

( n - 4 ) " ' (n—6)"' r

- * r Cn-2)"

~

- 2 - 3 (n—3) !•2-3 ( n - 5 ) a - 3 *-*-3 ("—7)-2-3-4-5

^ -

) " ' ^ ( n - 3 ) " ' ^ ( " - 5 ) " '

+ ••

3...(n — 2) * 1.2.3..(n-4).1.2 1.2.3...(n — 6).1.2.3.4 Praecedentibus, u t corollarium, a d j i c i p o t e s t , esse seriei

1 4. 2

x — 3

x

—

x

4, 5

x * 4 . 6

x

. . . summam Z ( S i n x 4» C o s x ) 4 . Z

(Sinx — Cosx).

• . * * I

1.2 (n—2)J

1.a.3...(n^8)J

Si Coefficienter C; C'; C,; CJ etc. eXpressionibus recurremibus definiri debeane, consi- dereturfunctioe

, quaeperxmultiplicata, antequamdifTerentiatiosuscipiatur, dabit xe

= x 4. x

4. i£_ ^{4« —— 4.} etc. . . .

1. 2 1.2. 3

e* 4« xe* = 1

2x ^{4. JL^_ 4. _i*.— 4. etc. . .} 1. 2 1. 2. 3

e

4. 3xe* ^{* x}

^e

⁼⁼ 1 * 2

X ^4. ^J^. ^4.

1. 2

4* ^x

^4* etc 1. 2. 3

3

- * —±— X

^4« etc.

1. 2 1. 2.3

e

4« 7xe

^{4< 6x}

^e

^{4« x}

^e

^{s s} 1 ^4« 2

X ^4.

sicqueporro; unde, siinZ, Z

CoefficientesC; C,; C'; CJ; etc.quicunquepositivias- sumantur

(Z + Z

) e

s= 1 * ^2"x 44 ^L

^{* ^—x}

^4. etc. . . .

1. 2. 3 1. 2

quare, si a , ß y? ^ x

. 1

1 ^{+ x +} 1. 2 + Ct + Ct *

designent Coefficientes positivos assumtos

1

' *

x

+ etc.l

1 ^4. 2" ^{x * x}

^{4. x}

^{* 5}

^{. 4,} 1. 2

Inde sequitur

1. 2. 3 1.2. 3.4 1.2. 3.4. 5 ^x

^4< etc.

j

3 =

y =

^ =

3'

^

1. 2 4".

1. 2. 3 5"

1. 2

ß-JL _

1.2 - JL _

^ 1.2 1. 2. 3. 4

, = _ J :

- j -

1.2.3.4-5

^

etc. • • • • • etc. • •

1.2. 3 1 1. 2. Ct 3

ß 1. 2. 3. 4

ct

1.

2. 3. 4

1, 2. 3. 4

Apparet, si ponatur d

(* ^ = i * 2

x * J _ x* — _ J — x

*

* . * etc.

1 K£KS i.2 1 - 2 . 3 1 . 2 . 3 . 4

quantitates C; C'; C,; C| etc. et obtineri aequatione Z *fr Z

=s d

(^

^e~

.

Ad valores «, y, ^ • • • calculandos inde ab n=si, differentiationecontinua com¬

mode uti licet. Quos valores calculatos usque ad n = 8 repraesentabit istud triangulum.

Litteris ß, y • • • numeri in columnis verticalibus subjacenres, diversos valores ipsarum, « , ß, y - - - , post primam, secundain, tertiam etc. differentiationemprodeuntes significa11t. Numefi, 1; <?; 3 ; etc. laeva manu designant ordinem differentiationis, a quo pendet valor ipsius n ; numeri, hypotheniisae inscripti, sunt factores, per quos si quilibet va!orum a, ß, y • • • m11ltiplicentur, evadunt diff'erentiae sei-ierum, quas numeri paraI- lelli cum hypothenusa progredientes constituunt.

Sic, idoneis artificiis adhibitis, ulterius progredi licebit. Niinc quidem, antequam finem investigationi i m p o n a m , dabo seriem

1 — a" 4« — ^ +

1. 2 1. 2. 3 1. 2. 3. 4

calculatam inde ab n = 0 usque n = 7

i6

i e i e Q e 9 , e 9 .

e 5 ° .

e

2^S *

2' *

1 ^{1 1}

1 . 2 i .

J

1 . 2 1 . 2. 3 i. 2- 3 - 4

X -

*

Jl —

_£_

*

1. 2 2. 3 i . 2. 3 . 4

Jl _

1.2 1 . 2. 3 i . 2. 3. 4

3" 4 ' 5 '

1 . 2 I . 2 . 3 x i . 3 . 3. 4

Jl —

1. 2 1. 2. 3 i . 2- 3- 4

1 . 2 . 3 . 4 . 5 6

1 . 2. 3. 4. 5

4« etc.

4> ^etc.

4> ^etc.

u _

4> ^etc.

1 . 2. 3. 4. 5

4> ^etc.

•$> etc.

I. 2 . 3 . 4 . 5 6^S

1 . 2. 3. 4. 5

6« - * c C . 1. 2. 3. 4. 5

- •J«

etc.

1 . 2 . 3. 4. 5

F i n i s .

T h e s e s

ad D i s p u t a n d u m p r o p o s i t a e .

Nulla datur lineae rectae definitio realie.

I I .

Differentiale est differentiae

uiedio arithfnetico omnium differentiae vaIorum assignabilium correspondens.

HL

Luminis reflexio est purum elasticitatis phaenomenon.

I V .

Amor prodigii est magni momenti in amplificandis scientii*.

V.

Est spatium vacuum in mundo.

VI.

Telluris et planetarum orbitas perpetuo coargui probabile est.

VII.

Quae sit vera

causa adhuc nescitur.

VIII.

Planetae raoventur in medio, quod eoruni motibus non resistit.

IX.

Telluris figura accurate cognosci nequit.

E m e n d a n d a .

P a g .

Errata.

i linea io. x n

I 2

5 5

7

I O . x n + 2

penult. abtinebitur 4 anteult. ( i — l )

n — 3

io abrarftpatur

— 8 e t 9 ubique a

— 8 e t 9 — ( x >

— g — * 3 - debit

_ 9 _ antepenult. difFerentione

— 1 4. neussarlo

— 1 0 — 6. parsunt

10 — 19.

_ 1 2

— 9. T

- 1 3

permlt.

Theses VI. coargui

C o r r i g e.

x n1

X"*4<2

obtinebitur

( l - l )³n - s ; ( I - l )²n - 3

terminum abrumpatur

a—

( x j v

tlabit

difFerentiatIone necessario

i-3-3^-5 6 "

1.2.3.4.5.6 ^{• x ' . ,}

coarctari