50 Mr, Homer on [Jcly, 



The mode of aggregating progressively is well known : to the 

 product of each numerator by the number which stands over it 

 is added the product of the next preceding numerator by the 

 number beneath it in the lowest line; the sum is the succeeding 

 numerator. The denominators are found in the same way. 



The parenthetic abbreviations are in accordance with the 

 principle of reduction already so often alluded to. 



6. Among diverging series, those which Euler has named 

 kypergeometrical, present, along with the difficulty of determin- 

 ing their sum, a paradox in regard of the equivalence between 

 the series and its sum, which is only to be solved by viewing 

 them in relation to recurring series. Thus, having 



1 + * 



1 +x 

 1 + 2* 



l+3jr 

 1 + 4 X + 2 «» 



= 1 -T o: +. 2 a;* - 4 x^ + 8 o;^ - 



= 1 ~a; + 2x^-6^3 + 20ar* - ....&c. 



we infer that making a; = 1, the series of fractions -, -, -, &c. 



converge towards the value of 1 — 1 + 2 — 6 + 24 — 120 + 720 

 — &c. whose sum has been, correctly in this view, determined 

 by Euler to be -5963473, &c. 



The mode of solution employed by that great analyst was 

 chiefly valuable for the new and useful artifices in integration, 

 which it ehcited from his fertile genius. See Lacroixs Diff. 

 and Int. Calc. English Edition, § 414, 218. The result was 

 verified through a widely different process by Dr. Hutton (see 

 his Tracts). Any solution whatever must, very probably, prove 

 sufficiently laborious ; but it appears to me that that which is 

 deduced from the principles detailed in this paper, is less operose 

 than either of the above, and leaves less obscurity about the 

 rationale of equivalence. 



Example Vfl. — The general hypergeometric series 



1 — mx+w.m-rr .a:* — m , m + r ,m + 2 r , x^ + ..... 

 compared with formulae (4, 5,) becomes 



i „, (11) 



1 + -— rx 



1+-— m + r . X 



1 + ; 2rx 



1 + — — m + 2 r . .r „ 



1 + ■ Srx 



1 + -r— 



1 + .... 



In the particular case already spoken of, m, r, x, are each = 1 ; 

 ivhdicc 



1 .^ 1 + 2 - 6 + 24 - 120 + 720 - ... . 



""1+1+1+1+1 + 1 + 1 + 1+1 + &c X12) 



The operations by which the value of this series has been 



