LEIBNITII ET BEItNOULLII, E p i s T. XXXIX. 221

positae, & a Tc & a me solute, potest per nostras methodos ul- 1696tenus reduci ad aliam , cujus indeterminate separari postsint. Vidca- Deccmb.

mus jam quid proveniendum sit ex hac generali Serie — 4-^—

a a.

a. (* + b.)(a 2 b) a.(a + b.) (a + zb.) (a-j- 3 b) ^ C '

( intelligo per a & b numeros quoscunque, ita ut ^, a -f- b , a -f2 b, a -j- 3 b &c. faciant progreslioncm quamcunque arithmeticam; j]

st + 2 b

a

faciamus ergo — +

a + b

x

&T-

dx

a

a —I

+

a.(a + b)

« + b — 1x

4 -

a.(a~{-b. ) (a -f- 2 b)»Tji-l

4- *_4-

T a. (a + b) ^

&c. =

r-

mi-

st + 5 b — 1

4- &c., & transposito x a *, & divisa aequatione per x 1

j r J* st + Z) a + 2 b

dy a—b _ x x , x

a-(a-\-b.)(a-{-2b)erit

ö — j

X dx

— H--— 4 -

a a.(a+b)

a. (a+b).(a-t- 2 b)

& C.

?=y id quod hanc suggerit aequationem dy=yx 1 dx -±-x a ~ l dx\qux sane ipsilsima est Fratris satis generaliter proposita; unde prae-ter spem incidi in modum solvendi hanc aequationem per Seriemsimplicislimam. quam forsan Frater non ita facile reperiret, si sol-licitaretur propriam suam Kquationem, vel saltem hanc per Seriemsolvere. Sumamus jam aliud exemplum ; ubi proveniat xquatiodifferentialis secundi gradus. Quaeritur summa hujus Seriei ,

I , X . I . I „ X . X X

4- -- 4-L_

1.4 ^ 1.4.9

X

1.4. 9

+-

I- 4 - 9-4

+ — +1.4. 3

deditur 1 +

H-- &c. ponatur — + --4.

1.4.9. 16 r x 1 1.4 1

+ —--- &c. = y , differentiando fiet 1 4~ —■- 4—

1.4.9.16 J 1.2 1.4.3

= multiplicetur per x, & erit -y- 4- ~~

&c.

X'

1.4. 9 4

x

4 - &c.

_ xdy

d x

v-3

.XX . x

-j--j-&c.

1.4 1.4.9

differentietur iterum, & ha-dxdy 4“ xddy

dx 1

ablato 1,

provenit tandem Series identica

dxdy 4“ xddy

~dF l

E e i

_ X .XX

1 1 IT4