534 Prof. W. Thomson on the Tidal Retardation 



at an acute angle to the line from the earth's centre to the moon. 

 One of the simplest ways of seeing the result is this : — First, by 

 the known conclusions as to the attractions of ellipsoids, or still 

 more easily by the consideration of the proper " spherical har- 

 monic"* (or Laplace's coefficient) of the second degree, we 

 see that an equipotential surface lying close to the bounding sur- 

 face of a nearly spherical homogeneous solid ellipsoid is approxi- 

 mately an ellipsoid with axes differing from one another by three- 

 fifths of the amounts of the differences of the corresponding axes of 

 the ellipsoidal boundary. From this it followsfthat a homogeneous 

 prolate spheroid of revolution attracts points outside it approxi- 

 mately as if its mass were collected in a uniform bar having its 

 ends in the foci of the equipotential spheroid. If, for example, a 

 globe of water of 21,000,000 feet radius (this being nearly enough 

 the earth's radius) be altered into a prolate spheroid with longest 

 radii exceeding the shortest radii by two feet, the equipotential 

 spheroid will have longest and shortest radii differing by § 

 of a foot. The foci of this latter will be at 7100 feet on each 

 side of the centre ; and therefore the resultant of gravitation be- 

 tween the supposed spheroid of water and external bodies will be 

 the same as if its whole mass were collected in a uniform bar of 

 14,200 feet length. But by a well-known proposition J, a uni- 

 form line F F' (a diagram is unnecessary) attracts a point M 

 in the line M K bisecting the angle F M F. Let CQbe a 

 perpendicular from C, the middle point of F' F, to this bisecting 

 line M K. If C M be 60 x 21 x 10 6 (the moon's distance), and 

 if the angle F C M be 45°, we find, by elementary geometry, 

 CQ=-02 of a foot (about \ inch). The mass of a globe of 

 water equal in bulk to the earth is l'l x 10 21 tons §. And, 

 the moon's mass being about -^ s of the earth's, the attrac- 

 tion of the moon on a ton at the earth's distance is ^p x ^-r^, 

 or ono of a ton force, if, for brevity, we call a ton force 



the ordinary terrestrial weight of a ton — that is to say, 



the amount of the earth's attraction on a ton at its surface. 



1*1 x 10 21 

 Hence the whole force of the moon on the earth is nm>n ^^^ , 



270,000 * 



or 4*1 x 10 15 tons force. If, then, the tidal disturbance were 



exactly what we have supposed, or if it were (however irre- 



* Thomson and Tait's f Natural Philosophy/ § 536 (4). 



t Ibid. §501 and § 480 (e). J Ibid. § 480 (b) and (a). 



§ In stating large masses, if English measures are used at all, the ton is 

 convenient^ because it is 1000 kilogrammes nearly enough for many prac- 

 tical purposes and rough estimates. It is 1016 - 047 kilogrammes ; so that 

 a ton diminished by about 1*6 per cent, would be just 1000 kilogrammes. 



