Remarks on Professor Wallaceh Reply to B. 299 



Putting this product equal to the assumed value (C) and 

 connparing the co-eflicients of the powers of x, Euler finds 



A=m+n 



m m-l m n n n—\ 

 B=y-'~"2~'^T T~'~r~2~' ^^ ^^ reduction 



m-\-n m-{-n — 1 



The values A, B are thus deduced in exactly the same 

 manner as Professor Wallace obtains the co-efficienis of s:, 

 z^, in Vol VII. page 279 ; and he also obtains the co-efficient 

 of z^ and the general value of the co-efficient of z^, by a 

 similar, but long and complicated operation, in pages 279, 

 280, 281, of the same volume. On the contrary, Euier 

 takes a much shorter and easier path. For, after remark- 

 ing that the co-efficients C, D, E, &c. may be found in 

 m and n, by continuing the multiplication, in the same manner 

 as A, B were found,* he observes that the calculation of these 

 co-efficients by this method is laborious and troublesome, 

 and the truth of this remark is abundantly proved by the 

 long operation of Professor Wallace. Euler then says, that 

 the process of multiplication by which the series (D) is 

 obtained renders it very evident that the co-efficients of x, 

 x^, x^, he or the quantities A, B, C. D, &;c. are definite 

 functions of m, n, which retain the same form, whatever be 

 the values of m, n,f and it is therefore only necessary to find 

 the values of A, B, C, Sic. in terms of m, n, in some simple 

 case, as for example when rn and n are integer positive num- 

 bers, and the same values of A, B, C, &fc may be immediate- 

 ly adopted for any fractional or surd values because the pro- 



*Itis stated by La Croix, pag-e 163, Comp. Elem. d'Alg-ebre that Seg-- 

 ner found the g-eneral term of the series A, B, C, &c. by this method of 

 continued multiplication in the Berlin Memoirs for 1777. This must 

 have been substantially the same as Mr. Stainville's. 



f Aslig-ht attention to the manner in which the series (D) is obtained 

 by the multiplication of the series/»i,/«, will render it evident that the 

 co-efficient of any term a; , is an integer function of the powers and pro- 

 difcts of m, n, containing' a definite number of terms of the form c. m , n ^, 

 e,ybeing- integer positive numbers not exceeding j:*, c being a numeri- 

 cal factor, and c, e.f, being independent of ?n, ??., will be the same whether 

 m, n, be integers or fractions, consequently the form to which the sum of 

 these co-efficients c. m' . n< is reduced must be the same for all values of 

 m, n, whether integers' fractions, or surds. 



