SIR W. THOMSON ON VORTEX MOTION. 233 



at different parts of the boundary of this area, in the infinitely small time dt, 

 contribute no increments or decrements to ff [a?dy dz\ as we see most easily by 

 first supposing S to be a surface everywhere convex outwards. Hence 



d 

 dt 



ffw* =JJ\r£ **] = 2 JT^ a?**] (6 > 



But if N denote the velocity with which the surface moves in the direction of its 

 outward normal at (<#, y, z), we have, in the preceding expression 



§ = Nsec a .... (7), 



if a be the inclination of the outward normal to OX. Hence 



V. = / / [ccN sec a dy dz\ . 



But the condition as to limits indicated by [ ] are clearly satisfied, if, d<t 

 denoting an element of the surface, such that 



dy dz = cos a d a , 



we simply take J~fd<r over the whole surface. Thus we have 



dJ P=ff*™> ■ • • ■ ^ 



42. In any case in which V is constant, this becomes 



Y §=ff*™' ■ ■ ■ ■ <»> 



If now the varying surface, S, is the boundary of a portion of the matter — fluid 

 or solid — of uniform density unity, with whose motions we are occupied, the 



^-component momentum of this portion is V -^ ; and, therefore, equation (8) is 



the required (§ 40) expression. 



43. The same formulas (7) and (8) are proved more shortly of course by the 

 regular analytical process given by Poisson* and Green! in dealing with such 

 subjects ; thus, in short. Let u, v, w be the components of velocity, of any matter, 

 compressible or incompressible, at any point (a>, y, z) within S ; and let c denote 



the value at this point of ~ + ~ + ^ , so that 



r dx dy dz ' 



=<-(!+$)■ (9 >- 



du /dv . dw 



dx 



We have, for the component momentum of the whole matter within S, if of unit 

 density at the instant considered, 



/ / / u dx dy dz = / / ux dy dz — / / / x j- dxdydz (10). 



* Theorie de la Chaleur, § 60. t Essay on Electricity and Magnetism. 



VOL. XXV. PART I. 3 O 



