300 PHILOSOPHICAL TRANSACTIONS. [aNNO 1758. 



similar triangles sdk, srd, if m denote the sine and n the cosine of the angle 

 SKD, viz. of the inclination of the satellite's orbit to the planet's equator, radius 

 being 1 ; then will dr = sd x n = -77— ? and sr = sd x w = — J^ ; 1 being 

 the total gravity of the satellite on its primary. Now because the force sr lies 

 in the plane of the satellite's orbit, it changes not the situation of that plane ; 

 indeed it accelerates or retards the motion of the revolving satellite ; but this 

 acceleration or retardation, because of the shortness of the time, does not 

 amount to any sensible quantity. The force dr perpendicular to the same 

 plane continually changes its situation, and generates the motion of the nodes, 

 defined in the following proposition. 



Prop. 2. To find the Motion of the Node arising from the aforesaid cause. 



By the motion of the node, in this prop. I mean the motion of the intersec- 

 tion of the planes of the planet's equator and of the satellite's orbit ; and the 

 satellite's orbit I suppose to be very nearly circular. Let s be the place of the 

 satellite in its orbit sn, whose centre is c, (fig. 9) ; sp an arc described with the 

 centre c perpendicular to the circle of the planet's equator fn ; sb an arc des- 

 cribed with the same centre perpendicular to the orbit sn ; and sb take the 

 lincola sr equal to double the space which the satellite can run through, when 

 impelled by the force dr determined in the preceding corollary, while it des- 

 cribes the small arc jbs in its own orbit ; through the points r, p, with the centre 

 c, describe the circle rpn cutting the equator in n, which will show the situation 

 of the satellite's orbit after that particle of time, the node n being changed to n. 

 Join sc, cn, and draw sh perpendicular to the line of the nodes cn, also ntk 

 perpendicular to rpn. 



Now since the lincolas sr, N7n, are as the sines of the arcs sp, sn, it will be 

 sp : sr : : sh : Nm ; then in the right-angled triangle ntww, it will be w- ; 1 : : 



vm : Nw ; hence by compos, of ratios sG x w : sr : : sh : nw = : and the 



sjb being given, nw or the motion of the node is as sr x sh. In the rightangled 

 spherical triangles sfn, the sine of the angle n, (that is, the angle of the 



inclination of the satellite's orbit to the planet's equator,) is to the sine of the 



k 

 arc SF, as radius is to the sine of the arc sn, that is, m : -j : : 1 : sh, and there 



k ' k . k 



fore - = m X sh ; therefore - is as sh. But the force sr is as - by the corol. 

 of the preceding prop, and therefore is as sh : hence sr x sh, and so nw also, 

 is as SH^ that is, the horary motion of the node, generated by the foregoing 

 force, is in the duplicate ratio of the satellite's distance from the node. And 

 because the sum of all the sh"^, in the whole periodical time of the planet, is 

 half the sum of all the sc**, therefore the periodic motion is the half of that 

 which, if the satellite were always in its greatest declination from the planet's 



