Equations of Motion. 



13 



Finally we get by annexing these quantities to the corre- 

 sponding equations for a perfect liquid: 



(li) 'hi ., . , , ., 



A — = rr—r sin'Ow—r 



.11(1 r '>/ 



cot^^ 



-cot^'-l- 



<l'n 2(hi 2// (Tn 

 <lr rilr r rdo' 

 (I'll 2 (III 



?•"' sin '0(l<s- r((<f 



Y- 



Ip or ^ . .. / (I'r , 4 (Ir , 2 r , 



= /• (-2 iir—r sin cosO ir — rl r 1 1 1- 



ot \ (Ir (Ir r 



'•rdo 



d'-r . 2 (III . di 



■ + 



+ 



Z- 



rdo' rdo rdo 

 dp 



cot — . 



+ 



(/■'/ 



2 cos odw' 



7* sin "'' rsin'Odc- r sin Ode J 



"ir 



= rsin I) 1-2 sin inr + 2 r cos ^ rw— 



I' r sin Od<f <)t 



v[ rs\n (-4sin o 1 f- sin ^ h 



ilr dr r rdo' 



., die 2ic sin 



3 cos (- 



rdo r 



dUv 



2 dii 2 cot odi 



r sin Od<f'- rsinOdf r sin Od<s 



Note I. 



For convenient reference the various equations are col- 

 lected from the different parts of the foregoing paper; and a 

 tabulated analysis of the terms appearing in the equations 

 is annexed. Special attention is given to the terms in the 

 equations referred to cylindrical and polar co-ordinates. 



EH 



< 



3 

 Pi 

 o 



I 



o 

 O 



< 



& 

 o 



< 

 H 



r 





S 



. Y 'Ip='!^ 



I'dy 'H 



Z- ^^^^ = ''"' 

 I'dz 'U 



dp _'hi !i.(d'ti (Vu (I'll 



iidx 'U ,"\d.r di/ dz' 



dp _'h- >i./d''t)(l''r d'r 



I'dy '>/ /'\djc' dif dz' 



•2 7 dj) _'hv .'x/d'w (t'w.d'wX 



