New Equation in Hydrodynamics. 173 



presents, as is known, a surface cutting at right angles the 

 directions of motion, which we will also call a surface of dis- 

 placement. 



From this Professor Challis deduces 



d-h d^ d'\> d'\> ^ ,- V 



by reasoning which I, like the illustrious analyst M. Bertrand, 

 am unable to appreciate. Hence is evident the passage from 

 (6.) to 



f-((^VH|-)V(f)>o.--- ('•) 



which is the new equation he contends for. 



In his reply. Professor Challis, contradicting his former 

 assertions, states, without further explanation, that the use of 

 equation (7.)» which he had already employed, is not abso- 

 lutely necessary for arriving at the equation (1 .). Thus falls to 

 the ground his chief argument for the necessity of a new 

 general equation. And in fact it is easy to convince oneself 

 that the equation (1.) is only an analytical transformation of 

 (2.). From equation (4-.) we have 



c?J/ d"!) . d-^ 



differentiating the first of these with respect to x, we obtain 



du cP^ dxdii 



dx dx^ dx dx' 

 Putting for the sake of brevity 



and observing that 



dx __ u _ V 



and thence 



d\ _^ dx y dx dx^ ^ dy dxdy ' dz dxdz 



dV d^l,d^^ d^^^ d^l; d^ 



dx~ R ' * R3 ' 



there results 

 du_udV V; 

 dx~\ dx ■*" R3 



» L dx'^ \dx / dx^ dx dy dxdy dx dz dxdz J ' 



