Equations of Motion. 7 



By equating and simplyfying, we obtain the third equation, 



Z -^ = r sin o — - + 2 sin »u- + 2 r cos vie. 



I>r sin Od<p ot 



Equation of Continuity. — Tiie increase in the element for 

 time dt is 



— >-'sin d<f dr dd; 

 dt 



the difference between inflow and outflow, in the direction of 



r, is 



in the direction of 0, 



sin d(f dr dO-^ 



d(rvf> sin 0) 



—^ r d<p do dr; 



rdo 



in the direction (p, 



— ^^ — ^ r' sin d<p do dr. 



r sin od<p 



Equating and dividing by r^d<p do dr, 



dp . ^, d{Hr^f>) . ^, , d (rv,o sin 0) d(r sin Ou-/>) . 



— sin 0= — ^- sin 0-] — i — --\ — ^^ ^— ^sin O; 



dt r\lr rdo rsinOdo 



or 



dfj . ,, d(iir'\") . (Z(r/'sin^) , d(ic,o) . 



— sin 0=^ ^sin 0+-^^ ^H 1_^ sin 0: 



dt r\ir do d<p 



or if fj is constant, 



dim-') . ^, , d(vsin 0) dto . , _ 



—5^ -s\nO-\--^ '--] sin ^^=0. 



r'^dr do d<p 



Collecting the equations obtained, we have 



X ^— = rv —r sin iv\ 



f'dr >H 



V —=r 1-2 uv—r sin cos uf, 



[>r do 81 



Z f^ = ?• Sin 1-2 sin mr-\-2 r cos vw, 



/>r sin Od^ of 



d(ur^) . „ , d(vsinO) <tw . 



—^ ^sin 0-\-—^ ^-f . — sin 0={j. 



r\lr do dip 



