Equations of the First Order. 339 



if this system is inconsistent no solution exists ; if it be consistent, and 

 the functions 



are not mutually independent, let it be replaced by the equivalent system 



/x=o,/ 2 =a, ,../»=a,/ n+1 =o,.../„ +s =o, 



where f v / 2 , . . *f n +s are mutually independent. There are now three 

 cases to be considered. 



I. If s be greater than r, then no solution exists. 



II. If s=r, we have n + r equations to find p v p it . . *Pn+r in terms of 

 x v x 2 , ... x n+r : if the values thus found are such as to make (ii) a perfect 

 differential, that is if the functions f v f 2 , . ..f n +r are such that for every 

 pair the condition (iii) is satisfied, then we have an integral of the form 



e=<p(v v x 2 , ...x n+r ) + b, (vi) 



containing the single arbitrary constant b. If the conditions are not 

 satisfied then there is no solution. 



III. If s be less than r, we have a system similar to the original one, 

 only containing s more equations. 



"We may therefore apply the above process to this system, and so either 

 demonstrate the non-existence of a solution, or find a complete primitive 

 of the form 



s=<p(x v x 2 , . . . x n+r , a v a 2 , . . . « } ._ s ) + 6, 



that is, an integral of the form (vi), or fall upon a new system analogous 

 to the given system (i), only containing more equations than either of the 

 previous systems. 



By continually repeating this process, it is seen that we must either 

 arrive at a solution or prove that a solution does not exist. 



We have now to consider the case in which the dependent variable z 

 is explicity involved in the proposed system, which is therefore present- 

 able in the form 



: \ (vii) 



f n (g, x v x 2 , . . . x n+r , p v p 2 , . . .p TO+J .)=°- J 

 Now let 



0(z, x v x 2 , . . . x n+r ) = (viii) 



be any relation between the primitive variables which satisfies the given 

 system (vii). Differentiating (viii) with respect to each of the n-\-r inde- 

 pendent variables x v x 2 , . . . x n+r , we have 



dx x dz dx n+r dz 



Hence, determining^, . . ,p n +r> and substituting in the proposed system 

 (vii), we get a system of n equations of which the type is 



