1897.] Differential Equation of the First Order. 281 



Hence the condition (u ly <£) = gives a linear homogeneous 

 equation connecting 



dcf> d(j> dcf) 



du 2 ' du 3 " ' dllzn-i ' 



the coefficients being functions of u ly u 2 ... u^i-i- The full number 

 of independent integrals of such an equation is In — 2 and u x is 

 one of them ; we may suppose without loss of generality that the 

 rest are u 2 , u 3 ... u 2n _ 2 . 



The condition (u. 2} <£) = 0, when <f> is a function of 



U 2 , U 3 . . . Uoyi—2 ) 



is similarly satisfied by In — 4 forms of <£ of which u 2 is one and 

 the rest may be taken to be u 3 , u 4 ... u 2n - 3 . 



By carrying on this process we get a series of n expressions 

 %bi, u 2 ... u n , functionally independent of each other and such that 



(/, u r ) — 0, (u r , M g ) == (r = lj 2 ... n\ s = 1, 2 . . . n). 



We are now to shew how the integrals of the equation /= 

 depend on the functions u 1} u 2 ... u n . 



§ 5. Let us write *— for the differential coefficient of u with 



respect to x i} account being taken of the dependence of z,p 1) p 2 ... p n 

 upon xi. 

 Thus 



du r du r du r ■?'=" dpj du r 



dxi dxi dz j=i dxi ' dpj ' 



and 



*'=" fa\. du s du s duA _, \,?s 9 Or, u 8 ) ( dpi _ dpf 



iA^ityi dx~i¥i)~ {Ur ' ' ~HPi>Pj)^j Z*i< 



= (r=l,2 ...n,s=l, 2 ...n) (4). 



Putting /in the place of u s , we have 



'S^¥ = 0(r=l,2...«) (5), 



i=i OXi dpi 



since /= by supposition. 



The system of %n(n + l) equations thus found is, in virtue of 

 the relation /= 0, an algebraical consequence of the system 



dz _ dpr _ dps 

 dx r ~^ r ' dx s dx r ' 



which contains the same number of equations. 



23—2 



