66-i TROCEEDINGS OF THE AMERICAN ACADEMY. 



form (R • Xq, R • Y^. Of all the vectors which have the u curves for 

 lines some are lamellar, for, if v is any functioD orthogonal to u, defined 

 by the equation 



9u 9v 9u 9v 



9x 9x 9y 9y 



so that the curves of the families u — c^^ v = Cn cut one another at right 



(9v 9v\ , . . T 

 — , — J has for its Imes 



the u curves, and it is lamellar, since 



9'v 9^v 



9x • d]f 9y ' 9x 



If ( Jl^), J^) which has the u curves for lines is lamellar, so is the vector 

 [ Xq • F{v)^ Yq • 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 (Xi, Yi) is a solenoidal vector which has the u curves for its lines, 

 the vector [Xi • F(u), Ti • 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 - \, is expressible as a function of u alone, where 



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

 defined by the functions 



w =/i (^^ y), ^' =A (^- y); 



and if 



U=i{x,y), V=',]{x,7/) 



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, 

 coplauar with z — ; it is not difficult to prove, by direct transforma- 



* Lfime, Le9on8 sur les coordonnees curvilignes, p. 31 ; Lemons sur les fonc- 

 tions inverses, p. 5 ; Soinoff Ziwet, Theoretisclie Mechanik, I. llo and 128. 



