228 Sir William Thomson on an Alleged Error * 



" observe that Laplace's numerical calculations of the heights of 

 " tides in certain latitudes, and his inferences as to the latitude 

 " where there is no tide &c, fall to the ground." When I first 

 read Airy's treatise ten years ago on board the ' Great Eastern/ 

 I could not assent to his correction of Laplace, but, on the con- 

 trary, satisfied myself that Laplace was quite right. Not having 

 the Mecanique Celeste at hand, I set the subject aside for a time, 

 intending to return to it for the second volume of e Thomson 

 and Tait's Natural Philosophy,' with the first volume of which 

 I was then occupied. 



4. My attention has recently been recalled to it by reading 

 in a volume of ■ Tidal Researches/ constituting an appendix 

 contributed by W. Ferrel to the ' United States Coasts Survey 

 Report for 1874/ the following passage referring to Laplace's 

 solution for the semidiurnal tides : — 



" The results show that the form which the surface of the 

 " sea assumes in this case differs very much from that of a prolate 

 " spheroid with its longer axis in the direction of the disturbing 

 " body, and that for certain depths of the ocean the tides at the 

 " equator are inverted, low water taking place under the attracting 

 " body. For great depths, however, the tides were found to be 

 " direct in all latitudes ; and even in the cases in which they 

 " are inverted at the equator they were found to be direct to- 

 " ward the poles, and consequently there is a latitude in such 

 " cases where there are no tides. Laplace, however, failed to in- 

 " terpret correctly in this case his own solution, so that the nume- 

 il rical results which he has given for different assumed depths of 

 " the ocean are erroneous ; but still the general results just stated 

 (( are readily seen from the solution. Having failed to see the in- 

 " determinate character of the problem, he adopted a singular and 

 " unwarranted principle for determining the value of a constant 

 " which is entirely arbitrary in the case of no friction, but which 

 " vanishes in the case of friction, however small. This oversight 

 " of Laplace and the indeterminateness of this constant were sub- 

 " sequently pointed out by Airy." The " singular and unwar- 

 ranted principle " thus referred to is in fact an exquisitely subtle 

 method by which it seems Laplace had determined a constant 

 which is not arbitrary in any case, and which cannot be more 

 than infinitesimally modified by infinitesimal friction. Ferrel 

 further extends to Laplace's integration for the " diurnal tide " 

 the objection of indeterminateness which Airy had raised only 

 against his integration for the semidiurnal ; and he follows 

 Airy in an integration (not given by Laplace) for the " long- 

 period tide," in which a false appearance of determinateness 

 (strangely inconsistent with the indeterminateness asserted of 

 the solutions for the semidiurnal and the diurnal) is produced 



