Prof. Challis on a Mathematical Theory of Tides. 19 



sponding to the increment ds of space in the direction of the mo- 

 tion, and r 1 , r" the principal radii of curvature of the surface of 

 displacement at the point ocyz. (See the proof of this equality 

 in the Philosophical Magazine for March 1850, p. 173, and in 

 the ' Principles of Mathematics/ p. 183.) 



It is known that there are instances of the motion of a fluid 

 mass in which either the whole of it moves as if it were solid, or 

 it may be conceived to consist of infinitesimal portions each of 

 which has the motion of a solid. In such cases r' and r" will 

 both be infinitely great, because the lines of motion in each ele- 

 ment will be parallels, and the surfaces of displacement will con- 



dV 

 sequently be planes. We shall thus have -j- = 0, or V a func- 

 tion of t along a given line of motion. 



But whenever r 1 and r" are not both infinite, it is evident 

 that the parts of a given element are changing their relative 

 positions^ and that consequently the motion is incompatible 

 with the solid state, and is distinctively that of a fluid. In all 

 these instances the surfaces of displacement are curved surfaces 

 of continuous curvature, or are made up of parts that are such, 

 and their general differential equation is udx + vdy + ivdz = 0. 

 Hence for motion which is proper to a fluid, the left-hand side 

 of that equation is integrable of itself. Since also, from the 

 foregoing argument, that differential quantity is always either 

 an exact differential or integrable by a factor, it follows that the 

 analytical circumstance of its being integrable by a factor indi- 

 cates that the motion of each infinitesimal portion is like that of 

 a solid. It may also be the case that the total motion is partly 

 of one kind and partly of the other ; but in such instances it is 

 possible to separate one from the other and treat them indepen- 

 dently. By these considerations I come to the general conclu- 

 sion that udx-\-vdy-{-wdz is an exact differential so far as it ap- 

 plies to motion which is peculiar to a fluid. This theorem, although 

 I had not recognized it before, is not inconsistent with the results 

 of my previous hydrodynamical researches. 



Putting (d(j)) for udx + vdy + wdz, the equation («) becomes 



dx* ^ dif + dz* U ' W 



at the same time the integration of the equation 



gives V = ^pj 7 . This result is to be regarded as an integral of 



the equation (/3), because, from what is shown above, the two 



C2 



