PEIRCE. — ON CERTAIN CLASSES OF VECTORS. 299 



where F{u) is some single-valued function : if F 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 ?< curves are straight lines parallel to the x axis, the 



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



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



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



equation of the form 



K = r ■ F{ic). (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 



h^ = 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^ must be of the form 



K = r' F(u) ■ V'(v), (25) 



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

 obtain the equation 



h^ = r • ip(v), (26) 



or 



^ (~) = 0- (27) 



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

 curves for lines, its curl is of the form 



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. 

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



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



