IN THE THEORY OF DIFFERENTIAL EQUATIONS. 531 



x,y,y, and write "bt(x, y) for y, Z(a,b) will become a function of C(x,y). This, we 

 know, will happen or not, according as Z x ^ ab C y - Z y ^ a jC x does or does not vanish. Now 

 since zr and - C^Cy are identical, this vanishes when Z I|o6 + Z yXa 6 . w vanishes, and this 

 is what 



Z a {A z + A y nr + A y , {isr x + sr y i tr) } + Z b \B X + B y -nr + B y , fa + -nr y . w)\ 



becomes, after substitution of nr for y in A x , &c. Now y" — ^ is the same as 



A, + A y y' + A y ,y"~ 0, and as B x + B y y' + B^y" = : 



so that, as y =sr satisfies y" = ^, the coefficients of Z a and Z h vanish ; while, the primitive 

 of 6' = \ employed not being singular, neither Z a nor Z b is infinite. It might be proved by 

 direct means that y = II is a consequence of 0=0, a = 0, b = A, from which therefore it 

 might be obtained by elimination. 



I have hardly a right, according to my own definitions, to make Z xtab mean all I have 

 made it mean in the preceding process. Strictly, I ought to have made the suffix, not x | a, b, 

 but a? | (a | x, y ="sr) (6 I x, y' = w), or the like. When it is desirable to make part only of 

 the implications prominent in the symbol, some general warning may be given, such as in 



* J x\a,b... 



I now apply the preceding to Lagrange's example, which has been repeated by all 



writers. 



<p (x, y, a,b) = -y + ^ax 2 + bx + a 2 + 6 2 . 



Let P = 16 (1 + x 2 ) y + x* - (8a? 3 + lfa) y - l6y' 2 , 

 Q = 6' 2 - 466' - 4a, 



<P* + < Psy'= - y'+ ax + b > <P„ + <t>b &' = (2^ + 2a ) + (• + 2b ) 6 ' 



cr (a?, y, a, 6) = a? 2 + 46a; - 4a. 

 4xy' + x 2 + -y/P w x 3 + 2.x + 4y 2 (a? + 26) 



4 (1 + x*) " 1 + x* (l + aj^-v/P a? 2 + 46a? - 4a ' 



4y'- a? 3 - xWP 1 a; 4 + 2a? 2 + 4a?y' (3? + 4a) 



4 (1 + a; 2 ) ' * 1 + a? 2 (1+ a^^/P a; 2 + 46a? - 4a ' 



2 (x + Zb) 2 



X=-b'+^/Q, X b ,> 



x 2 + 46a? — 4a ' 



F=a6' 2 -(2a6 + 6)6'+6 2 -a 2 +(6-a6') A /Q, Y *Jx + MY i™ + b ) 



v x 2 + 46a: - 4a 



/, » 4a ^' + a? 2 + -v/^ , • :r, 



y - x~ y 7 j? — ' sin g u ^ ar solutions, P = 0, 1 + ar = 0, 



4 (1 + a? ) 



2 (1 + 6' 2 ) 

 b" - a = b" - - - , (26 - 6' + -v/Q) 5 singular solution, Q = 0. 



