Equations of Motion. 11 



,^_ dp _'l^_ , ,n /lYcr-n fhi _ II (I'll 2 dv rPu 

 fxJr of r\dr'' rdr v' r'do' rdd dz' 



T^ dp '^r , ^ /// (Pv , 3 f/?? , d'-v , 2du , r/'r\ 



rrdo r,t ,o\ ,lr' dr rdO"- rdO dz] 



dp _ 'i/r fiUpir dw d'w d'lr 

 pdz of fAdr' rdr r'dff' tfz' 



The cause that contributes —rr' also contributes the rate 

 of displacement —2v normal to r sin d<p dr if rv is a vari- 

 able. Similarly there are rates, —2 sin ^ jo normal to rdO (fr 

 and —2 cos Oiv normal to rdO dr corresponding to —r s'\i\-Oic 

 and — ?• sin cosOw' respectively. The first forms a f)art of 

 the rate of displacement of n along r sin^'f/cr and the second 

 a part of the displacement of v in the same direction. In 

 taking account of the acceleration of ii along r and of v along 

 0, it must be remembered that if everything is constant there 



2/r 

 is an inflow equivalent to a rate of displacement of — ■ at 



r 



surface r'^ sin Od<f dO in the first case and of cot^v at surface 



r sin 0d<f dr in the second case. To the rate of change in \^ 



1u 

 along r and of r along 0, 2i" and — respectively must be added 



r 



as in cylindrical co-ordinates. If w is constant along /•, and 



cr, as in the foregoing cases, the rates of displacement at 



2?r 

 the initial surfaces are 2 sin ^ u', 2 cos ^ jy, and — + 2 coiOv re- 

 spectively. ^ 



Grouping the rates of displacement, we have 



. , , (fn , 2h , , „ . _ , 



( 1 ) 1 normal to ?•■ sin Od<f d0, 



(/>• r 



(2) ——2v normal to /• sh\0d<p d0, 



rd0 



(.'3) — I — 2 sin 0w normal to rd0 dr, 



r sin Od^ 



(4) r- \-2v normal to r' sin 0d(p d0, 



dr 



