in Laplace's Theory of the Tides, 235 



article (112), or equations (3) and (4) and (10) of §§ 5 and 13 

 above, constitute a complete and convergent numerical solution 

 of the problem of finding the semidiurnal tide in a polar basin, 

 or ocean continuous and equally deep from either pole to a shore 

 lying along any circle of latitude on the near side of the equator. 

 Laplace's result, as we have seen, does the same for a hemisphe- 

 rical sea from pole to equator. But for a sea extending from 

 either pole to a coast coinciding with a circle of latitude beyond 

 the equator another form of solution (still, however, with but 

 one arbitrary constant) must be sought*, because Laplace's 

 form [(3) of § 5 above] ceasing to converge when x (or the 

 sine of the polar distance 6) increases up to unity, fails to 

 provide for the continuous variation of 6 from zero to any 

 value exceeding \tt. And, further, the method suggested by 

 Airy, when extended to a complete solution of the differential 

 equation with its two arbitrary constantsf, completely solves the 

 problem of finding the semidiurnal tide in a zonal sea of equal 

 depth between coasts coinciding with any two parallels of latitude. 

 15. Returning to Laplace's solution for the whole earth 

 covered with water, we find in the Mecanique Celeste the nume- 

 rical results referred to by Airy (but not quoted, because of the 

 supposed error in the process by which they were obtained). They 

 are of exceedingly great interest (when we know them to be cor- 

 rect) ; and, in the circumstances, I may be permitted to quote them 

 here. They are obtained by working out numerically the pro- 

 cess indicated in §§ 5 and 6 above, for three different depths 



of the sea, > 79&Z 3 o m . 9 k ot the earth's radius. The values 



of e, or — corresponding to these depths, are 10, 2*5, 1'25 



7 

 respectively ; and Laplace finds for the solution [(3) § 5] in the 

 three cases as follows : — 



* It is to be found by using Laplace's first differential equation [the one 

 from which he derives (2) of § 5 above by putting 1 — fi i =ar i (Liv. iv. art. 10.)], 



(l-^)2^_ 2 [34-^-2Kl-^) 2 >=-8H(l-^), 



and satisfying it by the assumption 



a = A + A 1 /x+A 2 ^ 2 + &c; 



which, however, is a complete solution with two arbitrary constants, to be 

 reduced to one by the proper condition to make u — at one pole (say, when 



P= + l). 



t The general solution indicated in the preceding footnote suffices for 

 this purpose. 



