276 Integration of Differential Equations. 
not stop to give it, but shall refer for the process to the same vol- 
ume and page of Lacroix’s work that we have done at the com- 
mencement of this paper. 
We will now show how to find the integral of (2). ‘If we 
change b into bV —1, in (b), we shall have the complete value of 
y in the general case when p is positive and less than unity ; and 
if p=4, «=1, by making the same change in (e) we shall have 
the complete value of y in this case of the general integral. 
If p is positive and greater than unity, we must change 5 into 
6 —1 in Lacroix’s integral, and change —v? in ours into +72, 
then using A and B to denote two arbitrary constants, we get 
4 if dx(a* —2) ‘cos. buley/=1+B af ves doa +v? e 
cos. bu%v, (f), for the complete value of y. If p is negative and 
finite, we must change ~—2? into +-z? in Lacroix’s integral, and 
b into 6V —1 in ours, then using A and B for the arbitrary con- 
stants we shall have y= A ef. : dx(a? + x? ys cos.bute + Bu. ie 
ys dv(a? ~v?) *cos.bulv/ —1, (g), for the complete value of y- 
We will now give some applications of what has been done to 
differential equations which can be reduced to the form of (2). 
d*y Ady 
dart & de 
B*zx"y=0, (h), and Et + AS Brey=0, (7), in which x and 
y are the only variables, y being considered as a function of 7, 
© being considered as the independent variable whose differential 
dr is supposed to be constant, and e is supposed to be the base of 
hyperbolic logarithms ; if we change the independent variable 
from z to uw, we must not consider dr to be constant, but we may 
suppose du to be constant ; also in (h) and (7) we must suppose 
| d(°Y 
eee i vetted to pes 
dr? D: to = We shall now put u=2""", 
Suppose that we would integrate the equations 
n+2=m, and du= constant in (h), then since y is regarded as 4 ; 
fonctieneh and w of 2, we get by well known formula of 
dy 
. +e ap , du. im aa 
differentiation ies Se me z"-}, and | “ — 
