364 



GENERAL PROPERTIES OF CERTAIN PARTIAL DIFFERENTIAL EQUATIONS 



din i ft/> _|_ - _j_ ft. 



dx a "•" dx* "•" • • • • ~r dx„ — V«i) 



dxi dx 3 ' " 



+ ^7 = V"2, 



(6) 



<ffnl l ft|2 _|_ 



+ 





V"»- 



Between the three quantities f, m , f m „, £,, there exists for all values of?, m, n the 

 relation 



dt 



dtnl , d£i. 



"77^7 i j, ~r j^. — U 



dx 



dx„ 



(7) 



This relation holds throughout the three-dimensional space determined by the direc- 

 tions x t , x m , x n ; and as these are perfectly arbitrary, it is clear that a like relation 

 holds in each of the Euclidean spaces determined by the directions x taken in groups 

 of three. There does not appear to be any simple expression of this form connecting 

 all of the ^-functions. 



Consider for a moment the differential expression 



?< 1 c?.r 1 + ii 2 dx 2 + .... + u n dx n ; (a) 



the conditions that this shall be an exact differential are f 12 = & 3 = = £,,„_i = 0; 



or, as these may all be written, 



d d 



dXi' rfx 2 ' 

 Ifl, W a , . . 



d 



dx„ 



= 0. 



(8) 



If this quantity is an exact differential, say d(f>, the function <£ must by virtue of (5) 

 satisfy the differential equation 



If (a) can be made an exact differential by choosing a proper integrating factor, we 



must have 



U& + «j& + «*£,• = (9) 



for all values of i,j, k ; or, as these may be written, 



U l } U 2 > 



U„ 



= 0. 



(10) 



