Analytical Princip les of Hydrodynamics . 2 7 



this form of the equation (3) is derivable from Mr. Moon's 

 equation p =f{p, v) and the equation (1). From the value of p 

 we get by differentiation 



dp _df dp dfdv 

 dx dp dx dv dx 

 /being put for shortness' sake for f(p } v). Hence, since 



dx dy" z3 dx dx dt \dxj dx 2 ' 



fa 2 



it follows, by substituting — for -~, that 

 J ^ p dx' 



dx dp D dx 2 dv \p ) dx 2 



Consequently, by substituting in (1), 



dt 2 D\dp D dv \p))dx 2 



But the partial differential coefficients -~- and ~- are functions 



dp dv 



of p and v. Hence this equation may be put under the form 



and is therefore equivalent to the above-cited equation, 



<Py_ f (dy_ dy\d 2 y 

 dt 2 ~ J \dx dt/dx 2 ' 



Since jj = ^[-f\ the last equation may be assumed to be iden- 

 tical with the equation (3), which, as Mr. Earnshaw has justly 

 remarked, "can be made to coincide with any dynamical equa- 

 ls/ d 2 u 

 tion in which the ratio of -~ to -j— can be expressed in terms 



of d l" 

 dx 



It has thus, I think, been sufficiently proved that Mr. Moon's 



hydrodynamical researches are founded on differential equations 



which are really the same as those employed by Mr. Earnshaw, 



and differ only in the process of investigation and form of 



expression, and should consequently lead to the same results. 



This identity indicates that the two mathematicians have argued 



correctly, although by different processes, from the same princi- 



