290 Mr Dixon, On Lies Solution of a Partial 



when n = 3, k = 2 the equation f= must so far as p ly p 2 , P3 are 

 concerned satisfy the condition for a ruled surface. This is neces- 

 sary but not sufficient unless k = n. 



§ 14. The number of complete primitives is unlimited, but it 

 should be noticed that not every solution involving n arbitrary 

 constants is a complete primitive in the sense that all other 

 solutions can be deduced from it. 



It may be that an equation 



F(z, x u x 2 ...x n , a x ... a n ) = 

 is such that from it and the n derived equations 



*£+g-0(r-l,1....0 



two or more equations can be deduced in which the arbitrary 

 constants do not occur. Let these equations be 



/ 1= =0, / 2 = 0.../ m = 0. 



Then all the solutions derived from F = by the ordinary 

 method of variation of parameters will satisfy this system of m 

 equations, and therefore will not include all the solutions of any 

 one of the system, say f x = 0. 



For instance the equation 

 is a solution of 



Z = PiXi + p 2 X 2 + p 3 X 3 + ptfzPs 



containing three arbitrary constants, but since it satisfies the 

 equation p 3 = also it is not a complete primitive in the true 

 sense. 



§ 15. If any complete primitive of the equation f= is known, 

 the general solution of the auxiliary equation (/, <f>) = can be 

 deduced* as follows. 



Let F {z, x lt x 2 . . . x n , a^, a 2 . . . a n ) = 



be the complete primitive. Then from this equation and its n 

 derivatives expressions can be found for a lt a 2 ... a n in terms of 

 z, x x ...x n ,p x ... p n ; otherwise it is not properly a complete 

 primitive. Let these expressions be u x , u. 2 . . . u n respectively. 

 Then u lt u 2 ... u n are n of the 2n — 1 solutions of (/, $) = 0. 

 Another complete primitive can be found bv taking 

 a n = aA + a.J), + ...+ (^-Ai-i + K , 



* See Mayer, Math. Ann., Vol. vm. p. 311. 



