VOL. LXXII.] PHILOSOPHICAL TRANSACTIONS. 311 



Subtract this latter series from the former, and then 



« « + h . _r_ a + 2ft & C = 



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



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



'Ira - (r 4-1) 26 . (:r + 1) 6 - ra /a + 26 



— -75 X S -t- ; , 3 2r , x (r + j). 



Let r = 1 , and we have 

 _J! ^il 6 4- a -t-^- - &c... = {la-Ab) X s + iliriS. 



1.2.3 2.3.4^3.4.5 v ' ~ ^ 4 



If«=i, * = o jf7 ^ -rb+jXT- 181 = 2s-^; 



1 3 . 5 c 17 f- 



«=!.* = 2 >T7773 ~ 27775 + 3TT5 ~ &c = T ~ 6s - 



Prop. 3. — To find the sum of the infinite series , r + L 2 ™ + t 3r + L + 



^±^ + m + ' 2n , ^ ■ + &c. By a si- 



2r + 1 . 3r + 1 . 4r + 1 . 5r +. 1 ' 4r + 1 • 5r + 1 . 6r + 1 . Jr + 1 ' -^ 



milar process Mr. V. finds the sum of this infinite series will be generally ex- 



. , irm — (3 4- 2) n (3r + 1) n - inn rm + n 



pressed by - T X 8 + gj bVTFFl? 



Let r = 1 , and we have 



in . »i + 11 . m + 2« |B 4r« — 5ra , 7n—5m 



77273" + 3.4.5.6- + 5.6.7.8 + &C " ' " " = — 6~ X S + ~W~' 



Ifm=i, »= i, r72T 3 —+3T4^ti +6^771 + &c -*-- =l s -iV 

 ro = i, n= i, 777.771 + 37737S + TTtfTTs + &c - '• • = L 6~ 1 6 S; 



7 12 17 l 



m = 7, « = 5> nTTs" + J7 7.776 + 7767778 + &c ' * ' * = 1 s ; 



Cor. If ?z : m as 4r : 3r + 2, the sum of the series can be accurately had ; 

 let therefore n = 4r and m = 3r + 2, and we shall have 



3'" + 2 , 7r+ 2 , & _ _J_ 



l.r+1.2r+1.9r+1^2r+l.Sr+1.4r+1.5r+l ' ~ 2r . r + I' 



If r _ 1 1 4- ? — + -- ii_ + &c - i • 



11 r — ' ' 1 . 2 . 3 . 4 ' 3 . 4 . 5 . 6 ^ 5 . 6 . 7 . 8 ^ 0CC — 4 ' 



r O ! I 1 .J ? L. Xrp _L . 



r — ^ 1.3.5.7 ~ 5.7-9.11 r 9-11 • 13. 15 ~ — 96' 



,, 5 - H , 17 . . 1 



r ~ ' 1 .7 . 13 . iy "*" 13 . 19 .25 .31 "•" 25. 31. 37. 43 "T" — 336" 



Prop. 4. — To find the sum of the infinite series 



r + 1 . 2r + 1 . 3r + 1 . 4r + 1 



j "' + " I mj-j?n , & 



"T ",. + i . 4r + I . 5r 4- 1 . 6V + 1 ' 5r + 1 . 6> + 1 . 1r + 1 . Sr + 1 ^ 



By a like process also Mr. V. finds that this series is = 

 (5 r + 2) n - irm ( , 2m — (4?- + l) n Irm — (r — 2) « , 4rm — (2r — ] ) n 



6T? - X S -r- 6>4 -)- ,2. r s( r+ i) "•" 12r'(r + l).(2r+l) 



Cor. 1 . By a like process also it is found that 



a !±i + &c. = 



1 . r + 1 . 2r + 1 . or + 1 r + 1 . Vr + l . 3r + 1 . 4r + 1 ' 



4ra — (3r + 2) 26 . (3r+ 1)26 — 2m ra + 26 _ 2ro 4- 6 



3H X S -J- 3r4 ^ ^ + x) - - g r * # ^ r + !) _ ( 2r + ,) 



