123, 124] The Tidal Problem 123 



To examine the stability of the motion, we compare it with a slightly 



varied motion in which the boundary varies slightly from the ellipsoidal 

 shape. Let us assume at any instant that the boundary is 



(366) 



and that the velocity potential is < + M*, where <3> is still given by equa- 

 tion (365). 



Assuming \P" and -^ to be small, we find, from a consideration of the 

 normal velocity at a point on the boundary, 



, o/^9^ y W z m\ 



=2 - ^- +f - +- -^- (367), 



\a? dx b 2 dy c 2 dz J 



6 2 'by c 2 c)z 

 so that ^ is algebraically of the same degree as ty. 



Let us use P lt P 2 > ... to denote products of powers of a?, y, ^ say 



and suppose that values for M* and ^ are 



^ = ^^ + ^^2+ (368), 



y>-=q 1 P l +q 9 P a + (369), 



then, on substituting into equation (367) and equating coefficients, we obtain 



1=2^, etc (370), 



where 



The components of the total velocity v at a?, y, z are 



a a^ 



- x - etc., 

 a ox 



so that 



3^ c 9^\ 

 y-7r- + -^-5-), 

 ^9^ c d* J 



where u is the velocity when ^ = 0, given by equation (346). Just as in 

 119, the pressure can be made constant over the surface by satisfying 



4> + V + V b 



= - Trpabc B f/ - 2 + + -^^-l + ^ cons ....... (372), 



\0t 00 / 



where 6" is a new constant. 



We have already seen ( 120) that the motion is stable as regards ellip- 

 soidal displacement, so that we may suppose that M* and ty are free from 



