I 



coniahied in the Third Book of the Mecanique Celeste. 433 



mentioned by 9' and ot' ; 9' being the distance on the sphere's 

 surface, between the variable radius and a fixed axis or dia- 

 meter, and ct' the angle contained between 9' and a determi- 

 nate great circle passing through the same axis. When the 

 variable radius coincides with the line r, we shall suppose that 

 6', ■jsr' and y have the particular values 9, ot and j/; in conse- 

 quence of which there will result r = rt (1 + uy). 



Instead of referring the variable radius to an axis assumed 

 arbitrarily, we may ascertain its position with respect to the 

 line r. Let v{/ denote the arc between the variable radius and 

 the line r, and <J) the angle contained between ^^ and the former 

 arc 9 drawn between r and the fixed axis : then \J/ and ip will 

 determine the position of the variable radius, and j/' will be a 

 function of vj/ and $, as well as of 9' and ot'. The arcs 9, 9', -^ 

 are the three sides of a spherical triangle on the surface of the 

 sphere, and the angles opposite to 9' and -^ are ^ and ot — w'. 

 Hence we obtain, by the rules of spherical trigonometry, 



cos 9' = cos 9 cos ;{/ + sin 9 sin \|/ cos (fs, 



sin 9' sin (ot — ot') = sin 4; sin (fs, 



sin 9' cos (-CT — it') = sin 9 cos vj/ — cos 9 sin 4* cos $ ; 

 and from these expressions it readily follows that the three 

 rectangular co-ordinates, 



cos 9', sin 9' cos ct', sin 9' sin ■ct', 

 are linear functions of the three, 



cos \I/, sin vl/ cos ^, sin vf/ sin (p. 

 It appears, therefore, that if j/'be a rational and integral func- 

 tion of the first three co-ordinates, it may be transformed into 

 a like function of the other three. 



We now proceed to the fundamental demonstration of La- 

 place's method ; namelj', that of the equation which takes place 

 at the surface of the spheroid. Lety'denote the distance of 

 the point assumed on the surface from any molecule dm of 

 the spheroid ; then, supposing the attractive force to be as the 

 nl\\ power of the distance, let 



V =//" + ' dm 



the integral being extended to the whole mass of the sphe- 

 roid ; and V will be the function which, by its differentiation, 

 gives the attractive force of the spheroid upon the assumed 

 point in any required direction*. Now V consists of two 

 parts. The first part is relative to the sphere of which a is the 

 radius ; it is evidently a function of /•, which we shall denote 

 by A. The second part is relative to the stratum of matter 

 between the surfaces of the sphere and the spheroid ; and, on 



• Mecanique Celetle, livre 3"", (J 10. 

 Vol. 66. No. 332. Dec. 1825. 3 I account 



