1897.] Differential Equation of the First Order. 289 



But the relations (6) and (7) are now only n - k + 1 in number 

 and therefore fail to determine z, p 1 , ... p n in terms of x l ,...x n . 

 They may be taken as determining z, p k , p k+l ...p n in terms of 



x lt ... x n , p u ... p k _ 1 and it is then easily found that 



^=0 (r=l,2...k-l), 



Bp r 



dz _ 8 (/, <fr t , 4>, . . . <fr M ) , 9 (/, t , <ft 2 . . . <ft m ) 

 3a> r 8 (# r , p 1} ... p m ) d(z,p lt ... p m ) 



(r = l, 2 ...»). 



Thus the relations (6) and (7) yield an equation connecting 

 z, x lf ... x n and leave k — 1 of the quantities p 1} p 2 ••• p n un- 

 determined. 



If then we suppose pn/'a -'-Pk-i to have the values just found 



dz 



for - — (r = 1, 2 ... A; — 1) it will follow from the equation (13) that 

 ox r 



dz 



ST**' 



and in the same way 



Thus we may say generally that the equations (6) and (7) yield 

 at least one relation among z, x x ... x n , and when they yield only 

 one, that is when m^n — k+1, that one relation satisfies the 

 differential equation f=0. 



Exceptional cases may arise here, again corresponding to the 

 tac-locus and cusp-locus. In the one case all the minor determi- 

 nants of order n — k + 2 formed from the matrix (10) vanish, in the 

 other those of order n — k+1 formed from (11). 



§ 13. We are thus introduced to classes of differential equations 

 in which certain of the forms of solution are wanting. Lagrange's 

 linear equation is an extreme case, in which k = n. It is of course 

 the only case that can occur when n = 2. In general the class 

 includes equations formed by eliminating the arbitrary constants 

 and function from an equation of the form 



yjr(v 1} v 2 ...v k ) = 



where v 1} v. 2 ... are known functions of z, x x ... x n involving n — k 

 arbitrary constants, and yjr is an arbitrary function. The differential 

 equation is of such a form that it is satisfied by equating n — k + 1 

 of the quantities p v , p. z ... p n to linear functions of the rest, the 

 coefficients involving z, x x ... x n and n — k parameters. For instance 



