Integration of Differential Equations. 275 



the terms which involve e^, e^, &c. we get (by the exponential 



l-c .1-2? 1 



theorem,) w {a'^ — x-) = I + elog.M(a ^ — a;2y/, which when 

 substituted in (c) gives «/ = / dx{a^ —x") cos-fiw^fA+B-f 



Be\og.u{a^ —x^)ij, in which (although e is supposed to be indefi- 

 nitely small,) we may suppose that A+B is yet represented by 

 A', and eB by B', A^, B' being arbitrary finite quantities; .'. 



y=/ dx{a'-x^) cos.6w%i A'-f-B'log. M(a2 — a;^)?^ j;nowwhen 

 e is very small, {q being finite,) /is also very small, and {d) gives 



/ 

 2p = l—/, or ^= J by rejecting 2 in comparison with ^ ; hence 



when p = i (and q not indefinitely small) we get for the integral 



/a X 



cos.6m^^( A'+B'log.M(a2 _ x-yL (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) v/hen j9 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—7 



Again, if we put c=0, we get by {a)p = -^, and -g- = -p, 



and the particular value of y which we have found, becomes 



/»« 1^ 1 



y=tif dv(a^ —v^) '^'^ cos.bu^v, and if we put 6=- it will be- 



ra JJlI 1 l-q 



come y=u/ dv{a^ -v^) ^^ cos.-u'^v, or if we put -^ — =«, we 



1 Ai pa 



getl-g'=2zg'.-.g=2i+i'^"^2^-2=-2^p ^nay^uj ^dv 



_!_ d"y ~^'' 



{a^—v'^ycos.{^i-\- l)w2'+'i;, and(l) becomes ^j-7-!-a^2/w-'+' = 0, 



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 



