436 Prof. Challis on a Mathematical Theory of Oceanic Tides. 



itself. Now, however, I admit that it is absolutely necessary 

 that the equation (a) should be satisfied by the solution of 

 every problem which relates to motion of an incompressible fluid 

 for which udx + vdy -\-wdz is an exact differential. I therefore 

 fall back upon the first of the two solutions in the April Number, 

 which appears, for reasons I am about to adduce, to have been 

 too hastily rejected. 



The infinite wave might be accounted for in the same manner 

 as the continually increasing velocity of a pendulum, or an elastic 

 spring, which is subject to external action having the same pe- 

 riod as its own. Any disturbance of the waters of the supposed 

 ocean would generate waves which of themselves would be pro- 

 pagated with a velocity depending on the uniform depth ; 

 but the period of the tide-wave, or the rate of its propagation, 

 is determined by the mean relative periodic motion of the moon 

 about the earth. If, however, the depth of the ocean should be 

 such that the rate of propagation depending upon it is equal to 

 the rate of the moon's relative rotation, the waters might receive 

 continual accessions of velocity, just as in the case of the pen- 

 dulum or spring above mentioned. It may be presumed that 

 the infinite velocity given by the solution in question for a cer- 

 tain value of the ocean's depth is generated under theseconditions. 



This view is confirmed by actually calculating for the case of 

 an equatorial canal the depth H, which corresponds to a rate of 

 propagation equal to that of the moon's rotation about the earth 

 on the supposition that the rate = V^H. I find by this calcu- 

 lation that H is equal to 12*6 miles. This agrees closely with 

 the depth (12 miles) of an equatorial canal in which, according 

 to Mr. Airy's mathematical investigations, an unlimited tidal 

 elevation would be produced. (See Phil. Mag. for April, p. 267.) 



Since we found that in an unbounded ocean the infinite wave 

 is generated if the depth be 8*5 miles, the rate of propagation of 

 the lunar tide along the equator would appear to be greater in 

 such an ocean than in an equatorial canal in the ratio of ^12*6 

 to ^8*5; that is, of 1'22 to 1*00 nearly. This increment may 

 be due to the undulations which, according to the solution now 

 under consideration, take place in directions transverse to the 

 equator, in the same manner as, in the theory I have proposed of 

 the velocity of sound, the rate of propagation is augmented by 

 transverse vibrations, and very nearly in the same ratio. (See 

 an Article in the Phil. Mag. for June 1866, and ( Principles of 

 Mathematics and Physics/ pp. 214-224.) 



I have obtained the same solution by means of particular in- 

 tegrals of the three differential equations of which u, rvcosX, 

 and rw are the principal variables, the other variables being r, 6, 

 and X. In this method the relations between the arbitrary con- 



