in Laplace's Theory of the Tides. 229 



by the inadvertent omission of a constant*, the true value of 

 which is to be determined by a proper application of Laplace's 

 method. "With these results before me, I cannot wait two or 

 three years more for the second volume of ' Thomson and Tait's 

 Natural Philosophy ' to defend Laplace's process, but must 

 speak out on the subject without delay; and therefore I offer 

 the present article for publication in the Philosophical Maga- 

 zine, regretting that I did not do so ten years earlier. 



5. To Airy's statement of his case against Laplace^ quoted in 

 full below, I premise, by way of explanation : — 



I. The tide-generating force for the case in question (the 

 u semidiurnal tide ") is such that the equilibrium-tide height is 

 represented by the formula 



Hsin 2 0cos2</>, or H# 2 cos2<£, .... (1) 



where H is a constant, 6 is the colatitude of the place, or x the 

 cosine of the latitude, and <f> the hour-angle of the disturbing 

 body, which may be conveniently supposed to consist of moon 

 and anti-moon (two halves of the moon's mass) placed opposite 

 to one another at distances equal to the moon's mean distance 

 from the earth in a line kept always in a fixed direction through 

 the earth's centre and in the plane of the equator. 



II. Instead of H sin 2 or Ha?* in the equilibrium formula, 

 put a, so that it becomes acos2d>. This expresses the actual 

 tide-height if a be a function of the latitude fulfilling over the 

 whole surface the differential equation [Mec. Cel. Li v. iv. art. 10.) 



(l -*^£ —z - 3 (*-*- v * 4 ) a= - HE *> w 



where m denotes the ratio of centrifugal force to gravity at the 

 earth's equator (its value being actually about ^-g), r the earth's 

 radius, and y the depth of the sea, in the present case assumed 

 to be uniform all over the earth's surface, and but a small frac- 

 tion of the radius. 



III. Remark that the period of the disturbance thus investi- 

 gated is rigorously half the period of the earth's rotation — that 

 is to say, half the sidereal day. This supposition is no doubt 

 quite a close enough approximation for the solar semidiurnal 

 tide ; but it is certainly not practically close enough for the 

 lunar semidiurnal tide, its period exceeding, as it does, the half- 

 sidereal day by about -£ s of its value. 



IV. Remark also that if the earth's rotation is infinitely 

 slow, m is infinitely small, and the differential equation is satis- 



* See an article in the next (October) Number of the Phil. Mag., en- 

 titled " Note on the Oscillations of the First Species in Laplace's Theory 

 >f the Tides." 



