304 PROCEEDINGS OF THE AMERICAN ACADEMY. 



If ?t 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 



^2 — ^2 _ ^2^ ^45-j 



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



u = ±ax\r(yy) = \y V/ — a' — \ a" . log {y + V^' -«'') + X (^)' 

 60 that the complete integral of (45) is 



u= ± ax + hy Vy^ — a^ — ^ a^ .\og(j/ + V/ — a^) + 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. 



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