138 MR. GWYTHER OX THE LAGRAXGIAX FORM OF 



■vvhence the reqiiired form of tlie functions K^, K^, Kj is 

 given by 



or 



where P is a scalar, and Df- = o ; .'. !$ is a scalar function 

 of f , <f 1 f. only, and a takes the form 



VP + ^xVfx + -.V?. + -5Vfj. ... (4) 



Proceeding to find YVcr, fi'om this we get 



=Xa + l,i3+%y,saY, .... (5) 



where DtS = o, whence^ if the motion is irrotational, 

 the S's can be included in the vP under the above 

 hypotheses. 



Also Yv<T may be written Hi, ^ + H^, ^ + H^, :g ; 



xl ii ^11 



and since Yr~ ^^- denote the edges of the element,, the 



HS . 



components of "\ Vo" are — -^ — - &c. tunes the correspond- 

 ing edge; and if the fluid is incompressible this ratio 

 remains unaltered in time ; and the manner of variation, 

 if the fluid is compressible, is indicated by the formula. 



Also since D^S = 0, if Y V o" is ever = o it always was and 

 will be := o, and the strength of the section of the vortex 



* Conf. Mr. Hill's paper '" On some Properties of the Equations of Hy- 

 drodvnaraic-s,'" Q. J. of Math. Feb. 1S80. 



