266 Prof. Challis on the Mathematical Theory 



it will be found that C=— 0-0000070819^, the attracting body 

 being the moon. 



If r Y be the value of r where 0— //^ = 0, and r 2 its value where 



0—fit= — , and if X=0, we shall get by integrating the equation 



J=2C«(l-^) S in2(0-^), 



or by inference from the equation (s), 



By this formula the numerical value of r } -- r 2 is ^T12056 ft. 



Also the value of p 3 is found to be 1*02071 ft. Consequently 



the equation of the ocean-surface is 



y= a +1-02071 cos 2 X— 056028 cos 2 \ cos 2(6— fit) 

 = a + 1-58099 cos 2 X-l'12056 cos 2 \cos 2 {0—/it), 



the numerical coefficients being expressed in feet. Hence it fol- 

 lows that the tide consists of two parts — one of which is the 

 same for all longitudes, and varies as the square of the cosine of 

 the latitude, and the other varies as the square of the cosine of 

 the distance of any position from the point to which the moon 

 is vertical. So far as the form of the ocean-surface depends on 

 the latter part, it is that of an oblate spheroid the axis of which 

 always passes through the position of the moon, low water being 

 under the moon. 



Now, although this solution satisfies all the hydrodynamical 

 equations as well as the given conditions of the problem, there is 

 a circumstance to be taken into account which shows that it can- 

 not be the true solution. It will be seen that for a certain value 

 of b the denominator of the expression for C might become zero, 

 in which case this quantity and the vertical velocity would be 

 infinite. This inference is indicative of a breach of continuity 

 in the formula, to which there can be nothing corresponding 



in the movement of the fluid. The value of - which satisfies the 



a 



equation 



1 a 5 ~ G V 3flV 

 being found to be 0997879, it follows that 



a-b=ax 0-002121 = 8-47 miles. 



Accordingly, if the depth of the ocean exceed 8J miles, the maxi- 

 mum height of the tide passes through an infinite value, the 



