570 PHILOSOPHICAL TRANSACTIONS. [aNNO i78Q. 



&c.) X (aJ + Bb"^ + cb^ + nb* + &c.); then will the series ax + b^^ 4. ca^ + 



D^ +&C. — Aa7-t-j 2* T^l.2.3^ + 1.2.3.4a* "" ^ 1 . 2 . 3 . 4 . 5a> 



a?* + &c.; and the series \ -\- qx -^ rx"^ -\- so^ -\- tx* + &c= 



, B , 6CA — 2b* 2 , 18CAB — 8b' 3 , 36c*A* — 8b* 4 , o 



^^ A ' 1.2a* ' 1.2.3aJ * 1.2.3.4a* ' 



The terms of these 2 serieses can easily be deduced by the subsequent method. 

 Let Kaf~^ 4- La?""' + Ma,", be successive terms of the series Aa; + sar^ + ca:^ + 

 &c., and kV~' -f- L'a?""' successive terms of the series 1 -\- qx -\- rx^ -}• sx^ -|- 



, A , o 1.U Ml 2a*X BXK^ + 6CAK — 2b'k , , « X A X M — B X OJI. 



tx* + &c.; then will m = — -r , and l* = x . 



' ».(«— 1)XA* ' A* 



Cor. 1. Let b = O, and the '1 serieses Aa? + Ba?^ + ca?^ + Dar* + &c. and l + 

 qx + rx^ + &c. become respectively, 



^+M'^ + ^^s X r"'+ ..3.tl6.r X r.Xx^ + &c.,andl +f^^ 



If in these serieses for a be substituted 1, and for c be substituted — 

 -i-, there will result the serieses a; — -^ + J^ — &c., and 1 ~ -f 



&c. which srive the sine and cosine in terms of the arc x. 



1.2.3.4 ° 



Cor. 2. Let c = O, and the above-mentioned series ax + Ba;^ + &c. becomes 



*^+ r5 •'-' * - T:^.- X ?** - TT^frJTs X ^ -' + &c. The law of 

 this series is, first, that every 3d term vanishes; and 2dly, the signs of every 2 

 successive terms change alternately from + to — and — to + ; and lastly, the 



co-efEcient of the term x^ is ^ — - X -^^i and the series 1 -}- qx -\- rx^ -{- 



e 1 , . B 2b* o 2'b' 3 2'b* 4,0 T .1 • 



&c. becomes 1 + 7 ^ - ^iT*^ "" IT^iXT'^ " i.2.3.4a^ ^ + ^^- ^^ ^^is 

 series the signs of 3 successive terms alternately change from + to — and — to 



2" X b" 2* — ' X b" 



+ ; and the co-efficient of the term x^ is , ^ ■ or , ^ ^ according: as 



' 1.2.3. «a" 1 . 2 . 3 . . b a" o 



f? is divisible by 3 or not. 



2. Let a series 1 4- p-^^ + <^^^ + R'^^ + 6^* + ^a?^ + &c. be of such a formula, 

 that if in it for x be substituted a -\- b, there results a series 1 + p X (a + i) 

 4- a X (« + ^)' + R X (a + ^)' + s X (« + ^)' + &c. = (1 + pa + Qa^ 4. 

 kqS _|_ g^4 _^ gj^. ) X (1 + P^ + Gib^ + ^b' + sb^ 4- &c.) 4- (Aa 4" Ba^ -f ca^ 

 4_ Da* -j_ &c.) X (aZ? 4- B^^ 4- cP 4- dZ)* 4- &c.), then will the series ax 4- bx^ 

 4. c^ 4- D^* 4- &c. = A^ 4- B*^ 4- (^ - ^ + A X ^^^) x' + &c., and 

 the series 1 4- pa; 4- q^^ 4" i^-^ + &c. = 1 + p^ + 



A*+P% a . 2AB + PX (A*+P^) 3 4b^ + (A* + P^)' ' 4 , C^^ 



Let K^~* 4- L^" ~ 4- Mxf be successive terms of the series a^ + b^^ 4- cx^ + &c.. 

 and kV"' + !''<*''" ' + mV successive terms of the series I -{■ tx -\- qx"^ -\- bx^ 4- &c.; 



