338 Mr. E. J. Nanson on Differential 



the same time render the expression 



J> 1 dfc? 1 +i> a <&P a + . . . +J?n+rd#n+r (ii) 



a perfect differential, dz. 



First suppose the functions f v / a , . ,,f n are such that for every pair 

 the condition 



[/«./J=0 (Hi)* 



is identically satisfied. If we determine r new functions 



Jn + li Jn + 2i • ••fn+r 01 0G V X^ . . . W n + r , p v p 2 , . . . p n + r 



such that for every pair of the whole series f v f 2 , . ..f r the condition 

 (iii) is satisfied, then the values of p v p$> • . >Pn+r derived from the n equa- 

 tions (i) and the r equations 



Jn + li^ii x %i ' • • Mn+n Pv P& ' ' 'Pn+r) =:a p \ 



: I .... (iv) 



Jn+r\®ii ^2' * ' * ^n+rt Pv P%i • • •Pn+r) r=a n * 



where a v a 2 , . . . a r are any r arbitrary constants, will render the expres- 

 sion (ii) a perfect differential, and the value of z found by integration, 

 viz. 



Z=(j>(x v 06^ ... V n +r, & v Cl 2 , . ..a r ) + b, (y) 



b being a new arbitrary constant, will satisfy the given system (i), and will 

 be a complete primitive in the sense above defined. 



Now the determination of f n +i,fn+2, ■ -fn+r under the above condi- 

 tions is a part of the problem considered in Boole's ' Differential Equa- 

 tions,' Supplementary volume, p. 115 : the determination is there shown 

 to be possible, and a method is given for effecting it. Hence we see that 

 when the %n(n— 1) conditions [fi,fj]=® are satisfied, the proposed system 

 has a complete primitive. 



When the given system is a linear one, these conditions are identical 

 with those used by Boole, p. 81. 



Next suppose that the condition [/i,^-]=0 is not satisfied for every 

 pair of the functions f v / 2 , .../». Let the expressions of the form [fi,ff\ 

 which are not zero be denoted by <j> v 2 , 3 , . . . ; then it is plain that no 

 relation can be found which will satisfy the proposed system (i) without 

 at the same satisfying the system of equations 



1= O, 2 =O, 03 = 0,...; 



hence the required solution must be sought for as the most general solu- 

 tion of the system 



/i=0, /,=0, .:../»•=<>, fc=0, 2 =O, ^=0, ...; 



* Adopting the notation of Boole, Donkin, and others, the symbol [fi,fj] is used as 

 an abbreviation for the expression 



\dx k dp k dpk dx k ) ' 

 the summation extending from Jc=\ to Jc=n-\- r inclusive. 



