THE EQUATIONS OF FLUID-MOTION. 137 



the faces ^,, being due to the rate of change of the diflfer- 

 ential velocities just found. It will plainly be proportional 



to volume x V^— (Vv<'")V<Pi. 



II. 



The equation of motion is, on the generally adopted 

 theory, 



D,(r + V^'+ |^;VSvo-+JJ VV=0, . . (i) 



where V represents what would be written, in Cartesian, 



and therefore assumes each of these quantities perfect dif- 

 ferentials , and therefore § a function of jo only. 



We will first take /x^o. 



In any case cr can evidently be written in terms of the 

 normals to the three surfaces ip, determining the point at 

 which a- is the velocity. Thus 



a = K. Vf . + K, V «Pz + KjVf J. 

 On the previous assumptions K,, K^, and Kj must be of a 

 definite form, which we proceed to find thus : — 



+ KjS^V<P5JVf. + &c. + &c. 

 = (D,K, + So-^^-Wf.+&c. + &c. 

 =D,K.Vf. + D,K,Vfz + D<K3Vf5 + -'v(<^)S . (2) 



