342 Mr. E. J. Nanson on Differential 



a v a 2 , . . . a n -\ being arbitrary constants, and a n determined by the con- 

 ditions 



a 1 b l +a i b 2 + ... +a n b n + V< + a 2 2 + . .. +a n 2 =0. 



These values of p x . . .jp n , p give 



c 7 „ 



d<p = c^dla^ +.... + a n dx 2 + V 2a 2 . — ; 



/. (p=a 1 x l + . . .-\-a n x n + ^Za 2 Aogz + G; 

 whence the solution required is 



a x x x + . .. + a n x n + V2a 2 .log2-j-C=0, 

 which is equivalent to 



log *=c 1 a? 1 + . . . + c n a? n +D,- 

 D being arbitrary, and c x . . . c n connected by the equations 



Since there are « — 1 arbitrary constants, we have a "complete primitive" 

 as defined above. 



Example 2. Have the simultaneous equations 



\dxj ' ' ' \cfo n J 



-. 7 dz ,, dz 



any solution ? 



Proceeding as in the last example, we find 



^A = 1\" +i> 2 2 + • • •+ IV -^T 2 =0, 

 and the condition [/,»/ 2 ]=0 is not satisfied. Accordingly we write 



/, =[/,./,] =«*-o. 



From these we find 



and are thence led to the equivalent system, 



/ 2 ^& ] i? 1 +& 2 i> 2 +...+&»iv=o, 



Also it is easily seen that the functions 



f*r=Pv /«=JP2» ■••/»+ i=JP»-2 



