Remarks on Professor Wallace’s Reply to B. 299 
Putting this product. equal to the assumed value (C) and 
comparing t Phe co-eflicients of the powers of wt; Euler finds — 
serahdan 
mm-l1 mn 
i ae —~ eno tae , or by redueliaa 
sa mba 7 
R 
oO 
The values A, B are mee deduced in exactly the same 
manner as Professor Wallace obtains the co-efficients of z, 
z#,in Vol VII. pave 279; and he also obtains the co-efficient 
of 2? and the general value of the co-efficient of 2”, by a 
‘similar, but long and complicated operation, in pages 279, 
280, 281, of the same volume. On the contrary, Euler 
takes a much shorter and — trae a after remark- 
ing that the co-efficients C, D, . may ae found in 
m and n, by continuing the si ticalton in the same manner 
as A, B were found,* he observes that the calculation of these 
is operation of Professor Wallace. Euler then says, that 
w*, «2°, &c or the quantities A, Ww. We. are definite 
fun¢tions of m,n, which retain nae cen form, whatever be 
the values of m, n ml and it is therefore only necessary to find 
the values of A, B, C, &c. in terms o m,n, in some simple 
case, as for example when m and ” are integer positive num- 
bers, and the same values of 4, B, C, &e may be immediate- 
ly adopted for any fractional or surd values because the pro- 
*Itis stated by La Croix, page 163, Comp. Elém. bg mg i: that Seg- 
ner found the general term of the series A, c. by this method of 
continued multiplication in the Berlin emoirs for 1777. This must 
have been substantially the same as Mr. Stainville’s. 
+A slight attention to the eae? in which the series (D) is obtained 
by the ee of the series fm, fn, will poet it evident that the 
foo me teem x,isan ee function of the powers and pro- 
duets of m J of terms of the nia c.m, 
e, f be ing in ieiree positive numbers not exceeding p, c being a nu umeri- 
cal factor, and ¢, ¢ f, being independent of mm, x, will be the oA iibtbes 
m,n, be integers or young consequently the form to which the sum of 
these co-efficients c. m. reduced must be the same for all values of 
m, n, whether integers: ind or surds. 
