POINTS OF THE INTEGRAL CALCULUS. [123] 



by which any number of modes of transformation may be effected, no one of which alters the 

 order of the equation. 



In order to produce a system such as 

 X = (p (x,y,y), Y = f(x,y,y), Y' = ^,(x,y,y'), Y" = ^„(x,y,y',y"), &c, &c, 

 in which Y {n) shall be expressed in terms of x,y, ... y {n) , and nothing higher, it is the necessary 

 and sufficient condition that, X and Y being constant, X = (p and Y=\Jy shall be the two 

 primitives of the first order belonging to one equation of the second order. For Y ', or 

 ^x+^fy-y + </vy" divided by (p x + (p y . y + <p y . . y", cannot be independent of y", unless in 

 the form 



%l f.." ^ i'yy'+i'A . („" + <P</-y'+<p *\ 



„ m &t f ±dL±±\..[f. 



the two last members be identical : in which case X = \^ and Y = <p are primitives of the 

 equation of the second order obtained by making those members vanish. Such a connexion 

 of <p and >|/ being chosen, we have 



and if we can integrate / (X, Y, Y, &c.) = we can then integrate f((p, y]s, ^/-(p^ &c.) - 0, 

 by eliminating X and y between X — (p, Y = \|/, Y' = \|/y : (p^. And if we can integrate 

 the second equation, we integrate the first by eliminating a? between X = (p, Y = \j/. For 

 instance, if we choose X= x + y+y, we find, from l+y'+y"=0, that we may have 

 F" = e* (l + j/'), whence Y' = e x , Y" = e* : (l + y + y")> &c. Hence the equations 



/ {a + y + y', e* (l + y'), t", \' $ he.} = 0, and f(X, Y, Y, ¥", he.) = 



i + y + y 



have their primitives connected by the preceding rule with 



X=x+y + y', Y= e *(l+y') T - e*. 



The case previously given is that in which the two primitives chosen of y" = are X = y', 



Y = ivy' — y. A great many remarks will suggest themselves, in extension of those commonly 

 made on the equation xy — y = fy. Passing over these, I observe that if F (x, y, X, Y) = be 

 the common primitive of X = <p, Y = \^, then the system may be inverted as follows. Let X 

 and Y be the variables, and x and y the constants. Then the same three equations X = <p, 



Y = \J/, Y' = \|/^ : <pg contain the differential equations of the first order belonging to this 

 inverted system. If we eliminate y between these three equations, we have them ; and if we 

 thus get 



x = 4> (X, Y, Y), y = * {X, Y, Y'), we shall have y - * r : 0> r .. 

 Thus in the instance above, we have 



x = logY', y =X-logY--^+l, y ' = — - i = * r . : <b r . 



and, x and y being constants, the first two equations are two primitives of one equation of the 

 second order, namely Y" = 0. The original primitive is Y = e* X + (y + x — l) e". 



40—2 



