ON PHYSICAL LINES OF FORCE. 475 



Remembering that the surface of the vortex is a closed one, so that 



ZnxdS = ?,mxdS = ZmydS = tmzdS = 0, 

 and 'S,mydS=tnzdS= V, 



we find tPudS=(^/3-^y) V ........................ (49), 



and the whole work done on the vortex in unit of time will be 



rUf i 



^ = -~ 2 (Pu + Qv + Rw) dS 

 at 477 



1 ( fdQ dR\^ fdR dP\ (dP dQ\\ v . 



= - lai-r- -- i-) + P -7 --- T~ +7 j -- -rlf * ...... (50). 



<iir\ \dz dyj \dx dz/ \dy / dx/j 



PROP. VIII. To find the relations between the alterations of motion of the 

 vortices, and the forces P, Q, R which they exert on the layer of particles 

 between them. 



Let V be the volume of a vortex, then by (46) its energy is 



V ......................... (51), 



dE 1 ,,/ da O df3 , dy\ , . 



and -jr = 7-/ A ^ a -77 + -+r -57) ..................... (52). 



at ir r \ at dt ' at) 



Comparing this value with that given in equation (50), we find 



dR da/dR dP cZ\ IdP , . 



--~~ ...... (53) - 



This equation being true for all values of a, ft, and y, first let ft and y 

 vanish, and divide by a. We find 



Q . ., , 

 Similarly, 



dQ dR da 



___ ^* ^ - ~~7 II 



dz dy dt 

 dR 



(54). 



dP dQ dy 



and -^ y 51 = IL -r- 



dy ax dt 



From these equations we may determine the relation between the alterations 

 of motion -7-, &c. and the forces exerted on the layers of particles between 



602 



