286 Prof. Challisow a neijo Eqiiation in Hydrodynamics. 

 and the above result is reduced to the following, 



du dv dw _ dY \t { ^ ^ \ 

 d^'^dy'^d^~'ds'^\V^ V)' 

 Consequently, by substituting in equation (5.), we finally obtain 

 de dY ^jdp ,x / 1 1\ ^ ,^, 



It will be evident, from an inspection of the foregoing rea- 

 soning, that equation (6.) is arrived at whether N be a func- 

 tion of ^ only, or a function of ^ and the coordinates; that is, 

 whether zid x + vd y + 'w d z be integrable /)er se, or integrable 

 by a factor. Hence this equation may be derived from the 

 equations (?«) and {n) of the Meca?iique Analytique (part ii. 

 sect, xii,, arts. 7 and 8), in which ud x + vdy + w dz is as- 

 sumed to be an exact differential. I have, in fact, deduced it 

 in this way, but it is needless to insert the mathematical rea- 

 soning here. It is, however, important to remark, that equa- 

 tion (6.) obtained by this process is proved to be true only for 

 rectilinear motions of the fluid particles, whilst the proof by 

 equation (3.) is inclusive of the other, and extends to curvi- 

 linear motions. I will endeavour to illustrate this remark. 



In the Transactions of the Cambridge Philosophical Society 

 (vol. v. part ii. p. 196), I have given a proof of equation (6.) 

 totally unlike that above, and as it will serve to confirm the 

 truth of equation (3.) I will briefly state it here. In whatever 

 manner fluid is in motion, we may conceive at a given instant 

 an unlimited number of surfaces to be drawn, cutting at right 

 angles the directions of motion of the particles through which 

 they pass. In two such surfaces separated by an indefinitely 

 small interval, let two rectangular elements be taken opposite 

 to each other, so that the lines joining the corresponding an- 

 gular points, when produced, are normals to both surfaces. 

 By the nature of curve surfaces, if the sides of the rectangles 

 be assumed to be in planes of greatest and least curvature, the 

 normals will meet two and two at distances equal to r and ?', 

 the greatest and least radii of curvature ; and if ni be the rect- 

 angular element of the inner surface, and 8 r be the interval 

 between the surfaces, the element of the other surface will be 



(r + 8 r) (/ + 8 r) 



m. ' '-^, -. 



rr 



Also, if V, q be the velocity and density at the inner surface, 

 and V, q' be the same for the outer surface, the velocity being 

 considered positive when it is directed from the former to the 

 latter, then the increment of matter between them in the time 

 It'is 



