294 REPORT — 1876. 



To facilitate comparisons between the various results of the harmonic ana- 

 lysis contained in this and the preceding Eeports, and to promote a complete 

 theoretical appreciation of them all, the following harmonic analj'sis of the 

 equilibrium tide will be found useful. A portion of it, that, namely, per- 

 taining to the mean semidiurnal, the declinational semidiurnal and the 

 elliptic semidiurnal constituents, was given in the Report for 1872 (§§ 48, 

 50). For the sake of clearness, an investigation of the equilibrium tide, 

 consisting chiefly of extracts from Thomson and Tait's ' Natui'al Philosophy,' 

 vol. i., is premised, as the first edition of that work is out of print, and the 

 second edition can scarcely appear until after the publication of this E-eport. 



Let E denote the earth's mass, M the mass of the moon or sun, D the dis- 

 tance between the centres of the two bodies, a the earth's radius. If we 



neglect tides depending on the fourth and higher powers of =|T (of which only 



one, Laplace's terdiurnal lunar tide, referred to in § 3 of the Committee's 

 Eeport for 1868, and again in § 5, 1872, can probably be sensible), the equi- 

 librium tide will not be altered by the following arbitrary but conveniently 

 symmetrical assumption. Imagine M to be divided into two lialves, and let 

 these be fixed at distances each equal to D on opposite sides of the earth in a 

 line through its centre. Then if r, B be polar coordinates of any point referred 

 to the earth's centre as origin, and the line joining the two disturbing bodies 

 as axis, the equation of an equipotential sui-face is [Thomson and Tait, 

 §§ 804-811] 



"7 + ^^^ V(I)--2/Dcos^+.-) + V(B- + 2.Dcose + r--) ^=^'^^^^- " ^'^^' 



J- 

 and as the first approximation for :j^ is very small, we have 



f +^[l + i^,(3cos=0-l)]=const (12); 



whence finally, if »-=a-|-M, u being infinitely small, 



«=i^^3(3co8^e-i) (13)- 



. . . Ma^ 



This is a sjiherical surface harmonic of the second order, and -py^ is one 



quarter of the ratio that the difference between the moon's attraction on the 

 nearest and furthest parts of the earth bears to terrestrial gravity. Hence 



" The fluid will be disturbed into a prolate ellipsoidal figure, with its 

 long axis in the line joining the two disturbing bodies, and with ellipticity 

 equal to | of the ratio which the diff'ercnce of attractions of one of the dis- 

 turbing bodies on the nearest and furthest points of the fluid surface bears to 

 the surface value of the attraction of the nucleus. If, for instance, we sup- 

 pose the moon to be divided into two halves, and these to be fixed on opposite 

 sides of the earth at distances each equal to the true moon's mean distance 



the ellipticity of the disturbed terrestrial level would be nr 



2 X 60 X 300000' 



12 000 000 ' ^""^ ^^^ ^^''■'^ difference of levels from highest to lowest would 

 be about 1|- feet. "We shall have much occasion to use this hypothesis in 

 vol. ii. in investigating the kinetic theory of the tides. 



