272 Prof. Challis on the Mathematical Theory 



also evident from the above value of r. At the same time, by 

 reason of the equation (/c), we have 



2fic^ ( 2b*\ 2/z % 



The expression for c x shows that the first term of this equation 

 vanishes if /z.=0; that the other vanishes in the same case may 

 be thus proved. Since 



Sma/jb 



(>-*H--S-¥^£»- 



4GR 3 ^ 



and from the foregoing supposition 

 Sma/bi _ f b 5 \ 



4GR 3 

 it follows that 



Smb 

 4GR 3 6 



" G (, 1+ a*)* 



Hence, since the value of c } shows that fjbc x is equal to jtx, 2 x a 



positive factor, the value of f- has this form also. It thus ap- 





 pears from the value of c 2 that not only is the foregoing equation 



verified when /£=0, but we have also c 2 = 0. Consequently 

 the expressions obtained for u 9 v, and w severally vanish, as 

 plainly should happen in the case of equilibrium. 



I proceed now to calculate the numerical values of the coeffi- 

 cients in the expressions for height of tide and the vertical and 

 horizontal velocities, the attracting body being supposed to be 

 the moon, and the depth of the ocean three miles. From the 

 formula which gives the value of c 2 we have 



3ma* / b 5 \( b\ 

 Cj af _ #>Y_ 4GR a V a 5 J\ bj 



Q 2 



Hence, since by previous calculation -p~- =1-02071 foot, 



v^Trrnr' *- % =0-003786, ^ =1-001386, the left-hand 

 311*4 a? b 



side of the above equation will be found to be equal to 0'000776 



foot. Accordingly the maximum difference of the height of the 



waters is 0-001552 ft., and the equation of the exterior surface is 



