386 Mr. Challis on the general Equations 



that is a function of r and t to obtain this integral, only 

 because it is not possible, as it seems, to find the complete 

 integral of equation (2). Happily in the case in which the 

 motion is in space of two dimensions, the complete integral can 

 be found, and we are able to shew that the same result is 

 arrived at, whether we suppose ^ to be a function of r and t, 

 or determine the forms of the arbitrary functions in the com- 

 plete integral, on the hypothesis that the origin and direction 

 of co-ordinates are not fixed. This Proposition, which is im- 

 portant to the present theory, I have proved in the Annals of 

 Philosophy, for August 1829. 



2. The integral of (2) obtained above, seems to be that 

 which is really useful for the solution of any proposed question; 

 and it may be questioned, if the complete integral could be ob- 

 tained, whether it would be serviceable in any other way than 

 in conducting to this. For, let us now fix the origin and direc- 

 tion of the axes of co-ordinates in space, and let a, /3, y, be the 

 co-ordinates of the point towards which or from which the 

 motion at the point whose co-ordinates are a?, ^, z, tends. Then 



d^ -F(t) 



dr (.r-«r + (z/-j8r + {z-7)" 



also dr = dx + ■- — - dy + -dz, 



r r r 



,,,d<t>=^^^\'^d.^^-:^dy + '-:^dz\, 



a complete differential of a function of x, y, z, whenever a, /3, 7, 

 may be considered constant while x, y, z, vary in an indefinitely 

 small degree. From what has been said above, this will always 

 be the case where the parts of the fluid move inter se, and 

 change their relative positions: but when the fluid moves in 



