300 PROCEEDINGS OP THE AMERICAN ACADEMY. 



Any vector of the form [0, h^ • F{y), 0] is lamellar. 



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

 must have 



K = H^'), (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 



we shall get the new equation 



K^l. (31) 



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



have 



A, = F(u) ' ^(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^ + 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 (xq, r^) the equations of these lines might be written in the 

 form 



r — r 



V =^ 







X — Xg 



and the equation of the orthogonal curves in the form 



«2 = (x — Xo)- + (;• — r^y : 



ill this case we should have 



h^=(l + v"-)/n, 



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



