of Oceanic Tides. 267 



directions of the tidal motions are reversed, and the low water 

 under the moon is changed to high water. Bat clearly these 

 conclusions are wholly inadmissible. 



I take occasion here to state that the above solution appears 

 to be of the same character as that given by the Astronomer 

 Royal in the ' Monthly Notices of the Royal Astronomical Society ' 

 (April 13, 1866), to which I adverted in my former article in the 

 January Number. He finds that under the moon it is low water 

 "unless the depth of the sea exceed 12 miles/' the denominator 

 of the expression for the tidal elevation vanishing for this parti- 

 cular depth. The difference between 12 miles and 8J miles 

 is probably to be accounted for by the circumstance that the 

 former depth applies to an " equatorial canal," whereas the other 

 was obtained without any such restriction. To find that the 

 tidal elevation would be indefinitely great if the ocean had a 

 depth so small as 12 miles compared with the earth's radius, 

 is a fatal difficulty, indicating failure of the reasoning, and not 

 in any degree got over by saying that this depth " far exceeds 

 any supposed real depth of the sea." 



When in consequence of the above-mentioned results I had 

 almost despaired of being able to discover the right method of 

 treating this problem, it occurred to me to adopt the process of 

 reasoning I now proceed to explain. On the principle announced 

 in the former article, that any general integral of the equation 

 (a) which can be obtained prior to the consideration of a given 



case of motion (such as the integral V = -^7/-) has, to a particular 



solution applying to given circumstances of the motion, the same 

 kind of relation as that of the general integral to the particular 

 solution of a differential equation between two variables, it appears 

 that such particular solution, instead of satisfying the equation 

 (a), will satisfy a variation of this equation obtained by giving 

 indefinitely small increments to the coordinates. The reason for 

 this assertion is, that the particular solution coincides with the 

 general integral at a given time only through an indefinitely small 

 space. In fact, unless this principle be true, it seems hardly 

 possible to account for the failure of the solution above tried, 

 which satisfied the equation («) together with all the given 

 conditions. 



For conducting the proposed course of reasoning it will now 

 be convenient to employ, in place of the equation (a), the equi- 

 valent one obtained by transforming the rectangular coordinates 

 Wj y, z into the polar coordinates r, 0, X. When this is done 

 according to the usual rules, the result is 



d*.r<f > _1 d*.r<j> 1 d 2 .r<f> UnX d . r<j> _ . 



dr* + r 2 cos 2 X d0* r 2 d\* r 2 d\ ' W 



