PEIRCE. — ON CERTAIN CLASSES OF VECTORS. 299 



where F(u) is some single-valued function : if V is of this form it is 

 solenoidal. It follows from this that of all the vectors symmetrical 

 about the x axis which have the u curves as lines an infinite number are 

 solenoidal. If the u curves are straight lines parallel to the x axis, the 

 tensor of Q is some function of r or else constant. 



(b) If Q is to be solenoidal and if its tensor is to be either constant or 



expressible in terms of u, the gradient of the function u must satisfy an 



equation of the form 



K = r ■ F{u). (23) 



If for the function u in this equation we substitute a new function w 



defined by the equation 



/du 

 W)' 



we shall get the simpler equation 



K = r. (24) 



It is to be noticed that w is constant on any line of constant u, and that 

 (23) and (24) may be said to define the same curves. 



(c) If Q is to be solenoidal and if its tensor is to be expressible in 

 terms of v only, h u must be of the form 



K = r • F(u) ■ f(v), (25) 



and if for u we substitute w in the manner indicated in (5) we shall 

 obtain the equation 



h w = r • !(,("), (26) 



or 



Mr) = °- (27) 



(d) If a solenoidal vector symmetrical about the x axis has the u 

 curves for lines, its curl is of the form 



|W-«--£(^)] - 



If, for instance, the; lines of such a vector are straight lines parallel to 

 the x axis, its curl is either constant or a function of r alone, 

 (f) If Q is to be lamellar, it must be of the form 



V=h v -F(v). (29) 



