440 Frof. Challis on the general Differential 



which, by putting for u, v } and w their expressions above, becomes 



Hence by the foregoing reasoning the differential of this quan- 

 tity with respect to space-variables is equal to zero ; that is, 



But by the equation applicable to the first surface of displace- 

 ment (dijr) = 0. Therefore also 



K?«@?+f +f)})-°- 



Now this cannot be the differential equation of a surface of which 

 the variables x, y, z are the coordinates at the time t, because 

 the equation (dtyr) = 0, with which this one is not identical, is 

 the differential equation of that surface. Consequently the 

 equation can be satisfied only by supposing that 



It would be satisfied, it is true, if the same quantity were equated 

 to a function of the time ; but as such function may be supposed 



to be included in -~ t the above form of the equation is suffici- 

 ently general. 



We have thus arrived at a third general hydrodynamical 

 equation, by means of which the unknown function X becomes 

 determinable. It may be remarked that although, for the sake 

 of the argument, (^) = was supposed to be the differential 

 equation of a particular surface of displacement, the generality 

 of the reasoning is not thereby affected, because that surface 

 might be any whatever. The course of the reasoning is, in fact, 

 precisely analogous to that by which the second general equation 

 is established, in the investigation of which the principle of con- 

 stancy of mass is affirmed of a selected elementary particle. I 

 consider the foregoing method of obtaining the equation (3) to 

 De somewhat more complete than that given under Prop. VI. in 

 the Philosophical Magazine for January 1851. 



8. If T7, -tt, -rr be substituted for X-^-, X-p-, \~- respect- 

 eft at at ax ay dz r 



ivcly in the equation (3), that equation will apply to a given 



particle, and the left-hand side will be the complete differential 



coefficient of yjr with respect to t. Thus we shall have 



(f) = 0,a„d*=C. 



