24 Prof. Challis on a Mathematical Theory of Tides. 

 Hence by partial differentiation of the above equation 

 'dp* 



-e) 



dr 

 dp' 



~du' , Smrfi 2 . . ,,j .. d*u' 



"■m 



dp y 



'■(%) 



du' 3mrV . „.. . „ ,,, ,. dV 



We have now to introduce the condition that a particle at 

 the upper surface of the fluid remains at the surface in suc- 

 cessive instants, so that for every such particle I ~\ — Q, because 



the pressure is that of the atmosphere, which is supposed to 

 be constant. Hence the left-hand sides of the three prece- 

 ding equations vanish for a superficial particle. Thus the first 

 of these equations becomes 



_ ~du' Smra 9 ^ . n/A . d 9 u' 

 0= — G-^- + -^~ cos 2 \sm2(6—pt) -^ 



and, being applicable to every point of the surface at all times, 

 gives the means of determining what function u 1 is of the va- 

 riables r, 6, X, and t. It must, however, be noticed that it is 

 not the general integral of this partial differential equation that 

 the solution of the problem demands, but a particular integral 

 of definite form, also that there cannot be more than one 

 such integral suitable to the given circumstances. It is evi- 

 dent that the integral required for the case of an unbounded 

 ocean will be obtained by supposing that 



u' = A<j>(r) cos 2 \sm2(0—fjLt), 



without adding an arbitrary function of 6 and X, for determining 

 which there are no conditions. For by substitution we have 



0=-GAf(r) + ^f+4A^(r), 



an equation proper for determining the form of the function 

 <j){r). In fact, putting for brevity q for —^—, it will be found 



