300 PROCEEDINGS OF THE AMERICAN ACADEMY. 



Any vector of the form [0, h v ' F(v), 0] is lamellar. 



(f) If Q is to be lamellar and if its tensor is to involve v only, we 



must have 



K = ijy(v), (30) 



and the v curves must form a system of parallel, curved or straight, lines 

 (for instance, a set of concentric circumferences) ; the u curves are, 

 therefore, straight lines. 



If for v in (30) we substitute a new function z such that 



_ r dv 



we shall get the new equation 



K = 1. (31) 



(</) If Q is to be lamellar while its tensor involves u only, we must 



have 



h v = F(u) • t(v), (32) 



and the substitution used in (/) leads to the condition 



K = F(u). (33) 



We may consider that (32) and (33) define the same systems of curves. 

 By making use of the so-called " Principle of Duality," and putting 



m = — 2 log (x 2 + r"), n = tan"i (r/x), (34) 



it is possible * to reduce (33) to the equivalent of Fourier's equation for 

 the linear flow of heat. 



If the v curves were a family of straight lines emanating from some 

 fixed point (x , r ) the equations of these lines might be written in the 



form 



r — r 



v — — — , 

 x x 



and the equation of the orthogonal curves in the form 



u- = (x — x ) 2 + (r — ?- ) 2 : 



in this case we should have 



h v =(l + v?)/u, 



* Peirce, These Proceedings, 38, p. 663. 



