SIB Prof. Challis on some Points relating to 



given by the received hydrodynamical equations employed in 

 the usual manner, — first, because those equations do not in- 

 clude the principle of continuity expressed by equation (4.); 

 and secondly, because in treating this problem, udx + vdy + "wdz 

 has vi'ithout reason been assumed to be an exact differential. 

 2. I proceed next to trace the consequences of introducing 

 into the general equations the condition that udx + vdy + 'wdz 

 is an exact differential. Let {d<^)=udx + vdy + 'wdz. Then 

 {d(^) =0 is the differential of the equation of a surface cutting 

 at right angles the directions of motion. Hence the value of 



— + — in equation (7.) may be expressed by means of the 



partial differential coefficients of (p. This expression being 

 substituted in (7.), and p being eliminated by means of (5.) and 

 (6.), the result is identical with the known equation (n) in the 

 Mecajiique Analytique (Part 2. Section XII. p. 344). I have 

 indicated this process for the purpose of remarking, that it 

 does not thence follow that equations (7.) and (w) are identical, 

 or that the former is a particular case of the latter. Both 

 equations are equally general. The essential difference be- 

 tween them is, that (7.) involves an expression of the condition 

 that the directions of motion are normals to surfaces of con- 

 tinued curvature ; whereas («) involves no expression of this 

 condition, being usually obtained by simply supposing (p to be 

 a certain function, the partial differential coefficients of which 

 with respect to x, y and z are respectively ^<, v and w. It is 

 not possible to pass from (w) to (7.) without introducing this 

 condition of continuity, and equation (7.) consequently sig- 

 nifies something more than equation (w). 



We have now to consider the change which equation (4.) 

 undergoes by supposing udx + vdy + 'wdz to be an exact dif- 

 ferential. It might at first sight be supposed that for this 

 purpose it is sufficient to put unity for A. That this would 

 be a false step is clear from the consideration, that there would 

 then be six equations and but five imknown quantities, without 

 any reasons for concluding that the equations would be con- 

 sistent with each other. In fact, it would be found on trial that 

 on this supposition they are inconsistent. The only legitimate 

 process is to trace the consequence of supposing wrfj^ + u^^ + Mxia 

 to be an exact differential, by reasoning generally according 

 to the rules of analysis. 



Between the functions (p and ^ we have the relation, that 

 (<i\J/) = and {d(p) = are both differential equations of the same 

 curve surface. But (df) =0 being the differential equation of 

 a curve surface, it is clear that (^d, F(<p))=0, or F'((p)(t/(p) =0, 



