Fundamental Equations for determining Motion of Fluids. 427 



These, however, are merely suppositions of continuity, neces- 

 sary to be made that the subject may admit of tlie appUcation 

 of mathematical calculation, and virtual)}' contained in the 

 supposition of the existence of normal surfaces. Not being 

 able to discover any objection that can be raised against the 

 principles and the reasoning which have conducted to equa- 

 tion (A.), I shall assume this equation to be true. 



Now the only general equations of hydrodynamics hitherto 

 recognised are the two following: — 



dq d.gu d.gv d.g'w_ 

 rt'^~liT^~dJ~'^~dr-'^' ' ' ' ^^'^ 





(2.) 



which are so well known that I need not explain the significa- 

 tion of the symbols. These equations are perfectly general in 

 their character, and if they are sufficient in number for the 

 general consideration of fluid motion, the equation (A.) must 

 either be identical with one of them, or must be derivable from 

 the two combined. It cannot be identical with (2.), because 

 it does not involve the impressed forces X, Y, Z. For the 

 same reason it cannot be deduced from a combination of (1.) 

 and (2.) ; and it is not identical with (1.) for the following 



reason : — In the differential coefficient ' ° - of equation (A.), 



the variation of coordinates is necessarily from one point to 

 another of a line of motion, while the differential coefficients 

 of equation (1.) are subject to no such limitation. 



It follows, therefore, that another general equation, inde- 

 pendent of the impressed forces, must exist, by the combina- 

 tion of which with equation (1.), equation (A.) may result. 

 The considerations we have gone through will guide us in the 



investigation of it. It is well known that if — be a factor 



which makes udx-\-vdy-\-liodz integrable, 



— dx ^ dy -\ dz — Q 



AAA 



is the general differential equation of surfaces normal to the 

 directions of motion. Let the integral of this equation be 

 vj/ {x^y, z, t) = 0. We thus have an equation embracing all 

 such surfaces at all times. It must, however, be observed 

 that the function ^ cannot have this comprehensive character 



