practical Instrument of Physical Research. 491 



Let p be the pressure, a the (vector) velocity, and v the 

 force potential at any point of a fluid. Let f dp j (density) be 

 put as usual =P. Let d/dt denote differentiation with regard 

 to time which follows the motion of matter. The fluid may 

 be finite or infinite, and there may be any possible kind of 

 discontinuity in the motion, ft denotes a surface integral 

 taken over the true boundary of the fluid (including the 

 surface at infinity if the fluid is infinite). \\ denotes a surface 

 integral taken over this true boundary, and also over both 

 sides of any surface of discontinuity, which is thus supposed 

 to bound the region on both sides. The surface to which jj 4 

 refers may not always contain the same fluid particles, since 

 for instance fissures may form leading to new parts of the 

 true boundary, or such fissures may close up. When a 

 fissure is in process of formation \\ b refers to the boundary 

 thus created from the instant when the portions of fluid 

 begin to move asunder. Let t be twice the (vector) spin ; 

 i. e. T=Yycr. Let n be the convergence ; i. e. n = $^o. Let 

 N be* the surface expansion; i.e. N=— SUVcr. Let u be 

 defined by equation (4) above. Then 



M. + P»=Ji<, + P) S n >v ,,,,,-{|f,/-ifi) + f,,«} 



This may be put also in the form 



4w(t7 + P— o*/2)= | (v + F)$~Uvyuds- {<r 2 l2)8TJvyuds 



The usual integral for an infinite irrotational continuously 

 moving fluid, 



can easily be shown to be a particular case. It may be 

 noticed that 



Y(TT—n<r= — Yyo-cr, 

 and therefore 



— Svm(V<7T — no) = S V«yo-cr. 



* It might be thought that in analogy to n it would be better to put 

 N = SUW, but by considering the analogy between ordinary convergence 

 and a contracting bubble it will be seen that the definition of N given is 

 perhaps better. It must be remembered that \Jv points from the fluid 

 bounded and therefore into the bubble. 



