234 Sir William Thomson on an Alleged Error 



Laplace, however, did not suffer himself to be led into wrong 

 action by this misconception, and he seems to have entirely for- 

 gotten it when he goes direct to the right result, without note or 

 comment, by the truly "singular" process referred to above. 



12. On the other hand, Airy, after having, in the passage 

 quoted in § 6 above, allowed the same misconception to fatally 

 influence his practical dealing with the solution, closes with a 

 perfectly correct statement which is sufficient to show the ground- 

 lessness of his objection to Laplace's result, and the untenability 

 of what he substitutes for it. This passage has not only the 

 merit of inconsistency with the article which precedes it, but it 

 also constitutes a very decided advance in the theory beyond 

 any thing that Laplace either did or suggested, and for both 

 reasons I am glad to quote it. [Airy, ( Tides and Waves/ art. 

 (113).] " If, using the more complete values of a that we have 

 "just found, we proceed to form the values of ft"', b, and u, we find 

 " that w* will contain a series of terms multiplied by the indeter- 

 " minate K 4 . We may determine K 4 so that, for a given value of 

 " 0, u shall =0; that is to say, so that, in a given latitude, the 

 " water shall have no north-and-south motion. We might there- 

 " fore suppose an east-and-west barrier (following a parallel of lati- 

 tude) to be erected in the. sea, and the investigation would still 

 " apply. Thus, then, we have a complete solution for a sea which 

 €< is bounded by a shore whose course is east and west." 



13. Now in fact Laplace's process by the continued fraction 

 is a particular case of the determination of K 4 thus suggested by 

 Airy, though one for which Airy's method fails through non- 

 convergence ; that is to say, the casein which the proposed east- 

 and-west barrier coincides with the equator. For as we have 



seen (§ 8), Laplace's determination makes ^z=0 when #=±7r, 



and therefore makes the north-and-south motion zero at the 

 equator, as is obvious from symmetry, or as we see from the 

 general expression [Laplace, Liv. iv. art. 3; or Airy, arts. (85), 

 (95)] 



u—- r-5-?i(-7H : — n a— 4H siu OcosO ) cos 26 . (10) 



4m snr 6 \au sin 6 J r v ' 



for the southward component of the displacement of the water 

 by the semidiurnal tide. 



14. By § 7 above we see that Laplace's solution (3), with K 4 

 left arbitrary, is convergent for all values of x< 1. Therefore 

 it is continuously convergent for all values of Q<\tt. Hence 

 Airy's article (113), with the formulae which he gives in his 



* This denotes the meridional component of the displacement of the 

 water in any part of the sea. 



