Mr EARNSHAW, ON FLUID MOTION. 213 



MOTION OF THREE DIMENSIONS. 



12. The differential equation for this case is 



d/(p + clfcp + clj'cp = 0, 

 and its integral is 



•^ = i^, \f{x -a)+g{>/-0) + h{z- 7), f, g, h, t] 

 + F,{-/{x-a)+gUj-(i)+h{z-y),f,g, h, t\ 

 + F, { f{x -a)-g{y-^)+J,{x- y), f, g, h, t} 

 + F. {/{X -a)+g(>J-fi)-h{z- 7), / g, h, t}, 



f, g, h being constants, subject only to the condition 



/■■' + g' + /r = 0. 



It appears that in this case each set of values of J', g, h furnishes 

 four independent arbitrary functions in the value of ^ ; and without 

 repeating the reasoning of Art. 8, we may at once state, that the 

 general value of Kp will consist of the sum of an infinite number of 

 such sets of A'alues as the one above exhibited : and the same physical 

 inferences may also be made here for three dimensions, as there for 

 two ; namely, that each set of values of f, g, li furnishes a distinct 

 wave-surface, which is transmitted independently of all the others. 



It may also be mentioned, that if these distinct wave-surfaces 

 should be so situated as to be geometrically describable according to 

 the same law, that is, if their equations be of the same fonn, and 

 differ only in the value of the parameters which enter into them, 

 and if those values are consecutive, then we are not to take the 

 separate surfaces, but the surface which touches them all, as that 

 form of the wave which in such a case is denoted by the general 

 integral. Instances of this process will be given afterwards, see Art. 28. 

 This property will enable us to deduce integrals of the equation of 

 continuity adapted to wave-surfaces of any proposed form. See Art. 29. 



13. By reasoning precisely similar to that employed in Art. 9, 

 we may arrive at the following results which are general. 



E E 2 



