Integration of Differential Equations. 275 
the terms which involve e?, e*, &c. we get (by the exponential 
‘ c 1- = rite & WM 
theorem,) u (a? — x?) ae 1+ elog.u(a? — x? yr, which when 
substituted in (c) gives y= f dz(a2 =)" ‘cos.bule(A+-B+ 
Belog.u(a? —x? 4 ), in which (although e is supposed to - indefi- 
nitely small,) we may suppose that A+B is yet represented by 
A’, and ¢B by BY, A’, B’ being arbitrary finite quantities; .-. 
emel i 
y=[“de(a?—2?f cosibule| A’ +B! log. u(a? —2*)*) now when 
é is very small, (gq being finite,) f is also very small, and (d) gives 
2p=1~/, or p=4 by rejecting 4 in comparison with 4; hence 
when p=$ (and qg not indefinitely small) we get for the integral 
of (1), (observing that p=4 gives c=1,) y= [ds(a* ~2)-3 
4 
cos.butr (Ate log.u(a? ~ x*)9 , (e), A’, B’ being the two arbi- 
trary constants which (1), an equation of the second order of 
differentials, requires that the complete value of y should have. 
We may observe that Lacroix’s integral will always satisfy (1) 
when p is positive and not indefinitely small, but it will not sat- 
isfy (1) when p is negative ; also our integral will always satisfy 
(1) when p is negative, (whether it is indefinitely small or not ;) 
but when p is positive and greater than unity it fails to satisfy it. 
. —1 1-g 
Again, if we put c=0, we get by (a) P= a and Oy =—-p, 
and the particular value of y which we have. found, becomes 
}-9 1 P F 
yauf™ dv(a? —v?) ** cos.bu%v, and if we put ee it will be- 
(P:R CELA, Yo ag 
come y=u/ dv(a? —v?) 74 cos. ie or if we put a xt we 
+ 0 . 
: 1 .v Ai f 
get 1- q=2ig -.q=a577) apd 2q -2=— a7 4) and y=u (@ 
pe aad d? 
(a? —y? \icos,(2i + 1)u2**'v, and (1) becomes es 
the value of y being a particular solution of it; it would now 
be easy after the manner of Lacroix to deduce (from what has 
here been done) the second class of the cases of integrability of 
the equation of Riccati; but as it is sufficiently obvious, we shall 
—49 
Zap, 2th — 
+a*yu?'t! =, 
