304 PROCEEDINGS OF THE AMERICAN ACADEMY. 



If u is a differentiable function of x and y, the partial derivations of 

 which with respect to these variables are p and q respectively, we may 

 use the usual notation and rewrite (24) in the form 



P 2 = f — f- ( 45 ) 



Equating each side* to a constant, (a 2 ), we get 



u = ± a x + t (y) = h y Vy 2 — a 2 — % a? . log (y + vV — « 2 ) + X ( x )> 

 so that the complete integral of (45) is 



u =±ax + hy V/ - a 2 -\a 2 . log (y +Vy 2 - a 2 ) + b, (46) 



and the general integral can be formed from this directly. 



A special solution gives straight lines parallel to the x axis as possible 

 lines of a solenoidal symmetrical vector the tensor of which is a function 

 only of the distance from the axis of symmetry. No solenoidal sym- 

 metrical vector the tensor of which involved x only could have these 

 lines. 



The Jefferson Physical Laboratory, 

 Cambridge, Mass. 



* Forsyth, Differential Equations, p. 310. 



