310 PHILOSOPHICAL TRANSACTIONS. [ANNO 1782. 



a a + b a+26 a + 3b , . . , P 



HTTi ~ rTTTsT+1 + ,>,• + .,. 3r + , - ,7- + ,,„•+! + &c - for lf we re ' 



duce two terms of this series into one, it will become 



«ar — b . Qra +- (2r — l ) b ra + (4r - 1) b 



l . ;• + 1 . 2r + 1 ~*~ Qr + 1 . 3r + 1.4; + l ' ir + \ . or + 1 . 6r + l ' 

 where the denominators being the same as in the given series, and the nume- 

 rators also in arithmetic progression, we have only to take a and b such quanti- 

 ties that the respective numerators may be also equal ; assume therefore Ira — 

 b = m, Ira + {ir — 1 ) b = m + n ; therefore, b = — , a =. " ■ "" ~ \ -\ which 



substituted for a and b in Lem. 2, gives — "' — — - -f- - — — — !!L±_!i 



° l . r + 1 . 1r + I 2r + 1 . 3r + l . 4r + 1 



, m + 2» , » _ 2n» - (r + 1) « , (2r + 1) n - o m 



"r 4r + 1 . br + 1 . 6r + 1 ~ r ~~ Jr 3 AS-)- — . 



Let r = 1 , and we have 



m , m + n m + 2a . , . 3« _ 2w 



If m =], 1.^,^3+^+^+ &C.... = L-2S; 

 m =l, n = 0, _L- i + _i-_ + r -i- 7+ &c....= s -i. 

 If »=!, „ = i,_J__ + __^_ T+ _ r _^__ + &c..= I l X s + A ; 



m=l, W =0,--i--+ — j^ + 2i . 26 . 31 + &C - * =2l ~To- 

 Cor. If 2r : r + 1 :: n : m, the sum of the series can be accurately found, 

 and will be equal to - ■- — . Let therefore m = r + 1, and then n = 2r, 



consequently -^—j + 2r + , ? >4r + T + h^Ti + &c = 27 ; 



which is also known from other principles. 

 Prop. 2. — To find the sum of 



r + 1 . 2r + 1 • 3c + 1 "" 3; + 1.4; + 1 . 5r + 1 "*" 5r + 1 . 6r + I . 7r + I T &C * 

 Proceeding much in the same manner as in prob. 1, Mr. V. finds that the sum 



of this infinite series is equal to 



(2r + I) re — drm , Srm — (.ir + \) n 2rm — (;• — 1) « 



o75 X S "I 4;-* " i SF.T+" l"~ 



Cor. In prop. 1 , substitute a for ra, and 2b for ?j, and we have 



a _ j « + 26 , a + U 



I . r + 1 " 2r + 1 ' 2r + I . 3r + 1 . ir + 1 "*" 4r + 1 . 5r + 1 . 6r +1 ' &C ' - 

 ra - (;• + 1) b (^r_+_ 1)6 - ra 

 75 X & -f 2H 



Also in this prop, substitute « + /' fur m, and 2i for n, and we have 



a + fe I a+3h U & c = k+ Vb-ra 



r + l . 'ir + 1 . 3r + 1 ~ 3r + 1 . 4r + I . or + 1 ~ r 3 A s -}- 



ra - (2r ± 1) ft . ra + ft 



2r> "*" 2r 2 x (r + l)' 



