664 PROCEEDINGS OP THE AMERICAN ACADEMY. 



form (R • X , i? • T ). Of all the vectors which have the u curves for 

 Hues some are lamellar, for, if v is any function orthogonal to u, defined 

 by the equation 



9u 9v . 9u 9v 

 9x 9x 9y 9y 



so that the curves of the families u ^= c x , v = c a cut one another at right 



angles, the vector which has the components ( ^-, ^r- } has for its lines 

 the u curves, and it is lamellar, since 



9 2 v 9*v 



9x • 9y 9y ' 9x' 



If (X , T ) which has the u curves for lines is lamellar, so is the vector 

 \_X • F(v), Y ' F(v)~\, where F represents any ordinary function ; and 

 no lamellar vector has the same lines unless it is of the form just given. 



If (X u T{) is a solenoidal vector which has the u curves for its lines, 

 the vector [X x ■ F(u), T x ' F(u)~\ has the same lines and is also sole- 

 noidal ; no solenoidal vector has these lines unless it can be written in 

 this form. It will soon appear that of all the vectors the lines of which 

 are the u curves, some are always solenoidal, but no vector which has 

 these curves for lines can be both solenoidal and lamellar, unless u 

 happens to satisfy Lame's condition for isothermal parameters,* that is, 



unless 2 is expressible as a function of u alone, where 

 n u 



If a set of orthogonal curvilinear coordinates in the xy plane be 

 defined by the functions 



u—fi (x, y), v =/ 2 (x, y) ; 

 and if 



U=£(x,y), V= v (x,y) 



represent the magnitudes, at the point (x, y), of the components, taken 

 in the directions in which u and v increase most rapidly, of a vector, Q, 

 coplanar with z=0; it is not difficult to prove, by direct transforma- 



* Lame, Lei^ons sur les coordonnees curvilignes, p. 31 ; Lefons sur les fonc- 

 tions inverses, p. 5; SomoffZiwet, Tlieoretische Meclianik, I. 113 and 128. 



