XXXV 



But in the theories of Laplace and Damoiseau, the coefficients of the 

 several inequalities are presented in an unreduced form, involving denomi- 

 nators and auxihary quantities, which it is assumed are to he calculated 

 numerically, and by means of them the final numerical values of the coeffi- 

 cients are obtained. In Plana's work, on the contrary, the coefficients are 

 presented completely developed in powers and products of e, e , y, m (the 

 excentricities of the two orbits, the tangent of the inclination, and the ratio 

 of the mean motions) wdth coefficients, which are, of course, absolutely 

 determinate numbers. 



The results so presented constitute, to the degree of approximation 

 preserved, a complete algebraical solution of the problem ; they are deter- 

 minate results, in no wise dependent for their truth on the convergency of 

 the series, and they are of course absolutely independent of the particular 

 process made use of for obtaining them. They are consequently results 

 obtainable by the adoption of the time as the independent variable ; and 

 this is, in fact, the course followed by Lubbock — viz., taking the time as 

 the independent variable, he obtains directly the expressions of the true lon- 

 gitude, latitude, and radius vector in terms of the time ; expressions strictly 

 comparable with those of Plana, and which, but for the different significa- 

 tions of the e and y in the two theories, would be identical therewith. The 

 advantage of Lubbock's method is its directness ; the expressions for the 

 solar coordinates are in both theories given in the first instance by the 

 elliptical theory in terms of the time, and in Lubbock's theory they are 

 used in that form ; whereas, in the theory of Clairaut, they have to be 

 transformed into functions of the true longitude of the moon, and the so 

 transformed expressions are used in the calculation of the time and the 

 radius vector and latitude of the moon in terms of the true longitude ; and 

 there is, finally, the laborious reversion of series whereby the co-ordinates 

 of the moon are expressed in terms of the time. 



The researches on the tides are not easily described. They consist in the 

 main of the application of the existing theory to masses of observation. 

 Sir J. Lubbock was the first who introduced to the fullest extent the plan 

 of consolidating the results of all the observations : Laplace took chiefly 

 those made at the times when the irregularity under investigation was near 

 its maximum. A clear account of Lubbock's mode of proceeding is given 

 by Mr. Airy in his article on the tides (§ 489-491) in the *Encyclo- 

 psedia Metropolitana.' We may mention that the man of business was in 

 this matter a valuable colleague of the man of science. Mr. Lubbock's 

 relations with the late Mr. Solly, Chairman of the London Dock Com- 

 pany, procured him access to the observations made at the docks through 

 a golden number of years (1808-1826). We should rather say, procured 

 him the knowledge of the existence of the observations. Mr. Solly, we 

 are sure from knowledge, would gladly have communicated his information 

 to any inquirer ; and the Board would have given hearty assent, But 



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