312 PHILOSOPHICAL TP.ANSACTIONS. [ANNO 1/82. 



Let r = 1 , and we have 



_ a _ a + b _j_ a + '2b „ 4a — 10/) 834 - 32a 

 1 . 2 : 3 . 4 3.3.4.5 '3.-4.5.6 — 3 X S ^ 36 ' 



If a = 1, b = 0, ! ! u ! 2 - &c. . = - s - - 



' 1.2.3.4 2.3.4.5 '3.4.5.6 3 9 ' 



„ a 7. i ' 1 , X o 2 13 



a = 3, s= l, J- &c. ... = - s — 



'1.2.4 2 . 3 . 5 ~ 3 . 4 . 3 36 



Cor. 2. If a : h as 3r + 2 : 2?-, the sum of the series can be accurately found; 



take .-. a = 3r + 2, and b = 1r, and we shall have 



3r + 2 5r+ 2 , „ 1 



&c....= 



1 . r + 1 . 2r + I . .S/- +1 r + 1 . 2r + 1 . 3r + 1 . 4r 4- 1 ' (»•+•). (2r+ I) 



If r = J ' 1.2.3.4 ~ 2T3 ."4T5 + sTTTST 6 ~ &C = 00 ; 



r — o ': __i? J t frr — • 



' 1.3.5.7 3. 5. 7. 9. 5. 7. 9. 11 — 15' 



_ „ '1 _I7_ , _2J__ _ & _ I 



' 1.4.7.10 4.7.10.13 "*" 7 .10.13.16 °" ' 28' 



Prop. 5. — To find the sum of the infinite series — — - — , ,' — — ■ — ■ + 



1 . r + 1 . 2r 4- 1 . 3r 4- 1 . 4r 4- 1 ' 



m 4- n m 4 2« 



2r + 1 . 3r + 1 . 4r + 1 . 5r + 1 . 0) • + 1 "~ 4r + 1 . br + 1 . 6r + 1 . ?;■ + 1 . Sr 4- 1 

 + &C. 



This series also in like manner is found equal to 



2rm — (2r 4- 1) n (4r + 1) n — 2rm firm + .','» 2/v» + /; 



6r b - X S ■+■ — --— 72. r 4 (r+ I) 24r».(r4- 1). (2r+l)' 

 Let r = 1 , and we have 



m m + n , m + 2>i , 2m — 3fl w . 25?i — \6m 



1.2.3.4.5 '3.4.5.6.7' 5.6.7.8.9' -6 ' 72 



Um=>, n = 0, _^__ + _-^_-+_-L ¥ _ } + &c....=I S -| ; 



«. = 1, »= 1, --[-- + ~ 5 -- 7 + -~h-+ kc - ■ ■ ■ =s-) s ' 



m=25, n= 16, --^_ + _-^ + - r ^-— + &e. . . . = I s . 

 Co? - . If 7< : m as 2r : 2/ + ] , the sum of the series can be accurately found ; 

 assume therefore n = 2r, ?« = 2r -|- 1, and we have 



1 . _J -L&r ' 



1 . »■ 4 1 . .;/• 4 l . ir + 1 2r 4- 1 . 3r + 1 .5/' + 1 .6r + l' "*~6>.(r+l).(2r+l) 



If — 1 _J— J- _1 4- _!— JU — 1 



'1.2.4.5 "*" 3.4.6.7 "^ 5 . 6 . 8 . 9 KC - ' 30' ' 



r — 3 ' _i_ ! j_ . ' _ 4- &c = — 



' 1 . 4 . 10 . 13 ' 7 . 10 . 16 . 19 ' 13. U) - 22 . 25 ' ' ' 504' 



Mr. V. concludes this part with pointing out a remarkable property of those 



series whose sum can be accurately found : viz. that when the number of factors 



in the denominator is even, the numerator is always equal to the sum of the 



two middle factors; and when the number of factors is odd, the numerator is 



equal to the middle factor, and consequently will take it out oi the denominator, 



