OF ARTS AND SCIENCES. 



133 



X, T, Z, of an external force arising from the superficial tension of 

 the liquid, and the impulse given to the drop. 



If we follow the notation of Poisson * and Helraholtz,t we shall 

 have for the general equations of internal motion of a liquid : — 



-rr 1 dp (fit I dii 1 du I dit 

 h dx dt ' (/x ' dy ' dz 



rl dp do , do t du I du 



— '- = — -4- u— -\- V— -+- w— 



h dy dt ^ dx ^ dy ^ 'I- 



rr 1 dp dw 1 dw i dw i 



Z -L =^ — + U — -4- V — H- 



k dz dt ^ dx * dy ' 



W- 



dz 

 dw 



I 



' dz 



dh \ dh I dh I dh Sh / ,\ 



— 4- u— -\- V— -\- w — = — (4) 

 dt~ dx~ dy~ dz U ^ ^ 



du I dv I dw f-^ 



— -U — -I- — =. (0) 

 dx^ dy^ dz ^ ^ 



In which p is the pressure in a liquid at the point x, y, z; X, Y, Z, 

 are the components of the external forces acting on a unit of mass; 

 and h is the density. When tlie variation of h is infinitely small, we 

 have Eq. (5). The forces X, T, Z, are considered to have a potential 

 V. So that 



X=ir, T^II. Z^.'-f Eq.(6) 



dx dy dz 



and the velocities u, v, w, a velocity potential qp. So that 



u 



d<i> 





dtp 

 dy 



IV. = ^ Eq. (7) 



dx' dv' dz 



or, udx -f- vay -f- wdz = d(f, 



and q) satisfying the equation 



dx-' ' dy' ' dz^ 



which is what equation (5) becomes under the conditions expressed 

 above. We must therefore have 



du 

 dy 



du 

 d~x' 



do 

 dz 



dw dw du T-i /o\ 



cV dx = d-z ^^^•('^) 



* Traitede Mechanique. 

 t Crelle's Journ., LV., 1858. 



