Equations of the First Order. 343 



satisfy the conditions 



Hence we take 



a v a 2 , . ..« w _ 2 being arbitrary constants, and a n -\, a n being determined 

 by the conditions 



a* +< + ...+ «« 2 =0, 



a JL 6 1 + a 2 & 2 + . .. + a n b n =0; 

 and we find 



d<p = a l cLv 1 + a 2 dx 2 + . . . + a n dx n , 

 and therefore 



= a x x x + a 2 x 2 + . . . + a n x n + 0. 



As this value of does not involve z, there can be no solution of the given 

 equations. But the work has shown that the simultaneous equations 



ctKt) 2+ ---<y-<s 2 =°> 



M 



have an " integral " 



containing n — 1 arbitrary constants ; and that the system of simultaneous 

 equations formed by (a) and 



d± _() 



has the same equation for a " complete primitive." 



Example 3. Find the nature of the solution of the simultaneous equa- 

 tions 



du /du \ du 

 dx \dy ) dz 



x^(^+v)-^-z=0, 



Let 



+y)- 



\cty ; r \dz J j dx J 



du du ^ du 



dx dy dz 1 



fi=xp(q+y)—r—z=:0, 



2 



Then we find 



Uvf-zl^ipx-iXv+r-y-z)- 



Accordingly the condition [/ 15 / 2 ]=0 requires that 



px— 1 = 0, 



or else that 



q+r-y-~z=zO. 



=o.} <» 



