672 PROCEEDINGS OP THE AMERICAN ACADEMY. 



If for r and t in (25) we substitute their values as obtained from (20) 

 and (21), we shall get the equation 



q'-2 r' — 2p' q' s' + p'" t' = 0. (26) 



and this is (8) in expanded form. 



If equation (18) or its equivalent (26) is satisfied, it is evident that 

 by choosing ^{u) at pleasure we may find an infinite number of solenoidal 

 vectors which have the u curves as lines and have tensors which involve 

 u only. 



A comparison of equations (9) and (18) shows that the condition that 

 the u curves be possible lines of a set of solenoidal vectors the tensors of 

 which involve u only, is the condition that the o curves be possible lines 

 of a set of lamellar vectors the tensors of which involve u only. 



If ^ is a vector potential function of a solenoidal vector which has the 

 u curves for lines, and a tensor expressible in terms of v, — h,^F'(^a) is a 

 function of y, and h^ must be expressible as the product of a function of 

 u and a function of v, that is, 



h.^f{u)-4>{")- (27) 



If for II in this differential equation we substitute w, defined by the 



C du 1 . , 



equation w =^ j ,. , we get the simpler equation 



9h„ 



K = c/>(^') or "^^ = 0. (28) 



It is to be noticed that w lias the same lines as u, and that (27) and 

 (28) define the same curves; the equations (11) and (2S) are evidently 

 equivalent. 



If u is such that a solenoidal vector, V, can be found wliich has the 



u curves for lines and a tensor expressible in terms of v, its x and y 



fV9oV 9v\ ,^ T ,, 



components are -i- • ^r- , ^- . tt" • Ii we denote these components 

 \ h^ dx h„ 9ijJ 



by X, T, every other solenoidal vector which has the same lines has 



components of the form X-\p(ii), T-{f/(u), and the vector is not a 



function of v alone unless the factor if/ (m) degenerates into a constant, 



and the vector is a simple multiple of V. 



A comparison of (10) and (27) shows that if the u curves are possible 



lines of a solenoidal vector the tensor of which is expressible in terms of 



