Prof. Challis on a nexv Equation in Hydrodynamics. 285 



Similar expressions having been obtained for — and — , it 



will be found, by adding the three together and paying regard 



to the formula for 1 — r, that 



r r 



\dx dy'^ dzJKdx^'^ dy^'^ dzV ~lU^-di 

 _ _(P_^ d^ ^ d^ d<f> d''<f> d4> d^(f> d<^ 

 dxdt'dx dy^' dt dy dt' dy '^ dz-^'di 



dzdt'dz^ dAdx"-'^ dy^^~djy yv^Vy 



which equation may be reduced as follows to one of a simpler 

 form. 



Since V^ = .^ + ..+.,= N.(^V ^ , ^J), ,^„,. 



Uo„(3.)gives^ = V»(tf,^;,g). Hence 



(l£,df,d£_± d<l>^ 

 dx^^ dy^ "^ «/ ^2 - V2 • -^• 



Also by multiplying equation (3.) by N ^, it will appear that 



at dx 



and this equation, by differentiating with respect to x, gives 

 d^ dj> _ r/2 cf) (l±^± d^/2n. dY du\ 

 dx''-' dt dxdt'dx y^' dt^\\ '7llc~ d^)' 



So "^ 'i^--^^ #=± ^/?i^ ^V dv\ 

 dy^' dt dydt'dy Y^' di^ \V "J^ ~ J]/)' 



and "^ "^ - -^1±- ^ _ ± df/^w dV dw\ 



dz^'dt dzdt'dz y^' dtAT'-dz'lur 



When the several values thus obtained are substituted in 

 the toregomg equation, the result is 



dn (Iv dw_u dY V dV iv dV /i i\ 



dx-^ dy-^ dz- Y' dx-^ V'Thi + Vdz ^^[7+y)' 

 If now the condition be introduced that the variation from 

 one pomt to another of space be in the line of motion, we shall 

 nave 



u _(lx v^^dy iv dz 

 Y'~ds Y~ts* V^ds' 



