﻿54 Dr. G. Green on Fluid Motion 



In these equations, 1? 2 , Z represent the terras depending 

 on the rotation of the axes themselves, being given by 

 equations of the type 



#i = — 2o) z v + 2<Djiv — w : y + Wy.z 1 + co x (o y i/ -f ov^s' — (co/ + &)/) 0. 



... (4) 



The function V(#, y, *) represents the gravitational potential 

 function. We have also 



D 3 ^ 9 j. „ ^ j. * «•> 



D7 = 5t + l( ^ + ^ + "5i- • • • (5) 



The equation of continuity of the fluid is then 



W + K^ + ^ + 9l)=°- • • • (b) 



In applying these equations we treat the atmosphere as a 

 perfect gas in which viscosity may be neglected. 



Circulation Theorem for Relative Motion. 



Consider now the theorem relating to the relative circula- 

 tion. We have 



j^ (u dx + vdy + w dz) = y- dx + jydy + -^ dz + d (i q 2 ), 



• • • (?) 



where q 2 = u 2 + v 2 + w 2 , the square of the resultant relative 

 velocity. By means of equations (1), (2), (3), the above 

 equation may be rewritten in the form : 



yr- (u dx + v dy + w dz) 



= -(0 L dx + 6 2 dy + s dz)- C ^-dY + d(iq 2 ). (8) 



We can now integrate each term of this equation along any 

 curve within the fluid from any point A to any point B. 

 This integration gives the result, 



Dl'B rs C dp 



n I (udx + vdy + wdz) =—] (#i dx -\-0 2 dy + O s dz) — 1 - J - 



-V B + V A + i?B 2 --kA 2 ; (9) 



and, if the integrations are applied to a closed curve 



