EQUATION AND ITS TRANSFORMATIONS. 
767 
whence 
A 0 = — 
P 
1 a 2 . 
-„A„ 
_x 92 0’ 
A*=- 
,i J_ A 3 a 
1 2 ^ 
A e =' 
A_— _ A 
®p_$. 22 A ^’ 
so that the solution corresponding to the root w? = —p is 
U = aHN I 
-1 o o 
J crar 
fra* 
6 >.6 
0?X 
p-i 22 ' (p-i)0-|) 2t2! (p_i)(^-f)(p-f) 213! 
^+&c. , 
where, as throughout this memoir, r! denotes 1.2.3 . . . r. 
Similarly, taking the root m— —p —1, the other solution is found to be 
X —ocJ )+] 
1 cc 2 x 2 1 cdx 4, 1 a e x 6 p 1 
'p + 1 ~2 f+ (l> + m> + W 2t2! + (P + l)(^ + l)(^ + D 2^3: J 
and, as U and V are independent, the complete integral of the differential equation is 
AU + BV, A and B being arbitrary constants. 
There is nothing in the form of these series to indicate that for any values of p the 
integral of the differential equation admits of being expressed in a finite form. They 
show, however, that if p — the half of an uneven integer (the case p>——\ alone 
excepted) the solution assumes a different form, viz. if, say, in U the terms after a 
certain point become infinite, the solution is of the form W-f-Vlogcas, W being a 
new series. This case is excluded in what follows; and throughout the memoir p is 
supposed not to be of the forms ±a(2'n + l). If, however, p is of either of these 
forms only certain of the series considered will involve infinite terms, and the relations 
connecting those series which do not involve infinite terms will still remain true. 
2. Transforming the differential equation (1) by assuming v=-e ax v, a substitution 
suggested by the form of the first member of the equation, we obtain the differential 
equation in v 
d~v , _ civ v(v+ 1) 
—+ 2a—= yw - 3 . L v. 
ax' 1 cix x A 
Putting as before 
v— %K r x m+r , 
we have 
(m -f- r -\-p) (m +r — p — 1) A,.-f- 2 (m -f -r — 1. )a A r-1 =0, 
whence 
m=—p or p-{-l. 
