﻿512 Mr. D. Vaughan on Secondary Planets 



dinates of any point on the surface of the satellite, the centre 

 being taken as the origin of the three planes of reference, sup- 

 posed to intersect in the axis of rotation, the radius of the orbit, 

 and the line of orbital motion. 



Let A, B, and C be the maximum values of a?, y, and z, or the 

 major, mean, and minor semiaxes of the figure of equilibrium. 

 The attractive force of the satellite at the given point is equal to 



ci m ^ : and this may be resolved into three components 



m +y +z* 



parallel to each axis and expressed by 



mk*x mtfy mk*z m 

 »» P ~ T' * v 1 / 



(x 2 + y*+z 2 )* (x* + y 2 + z^ {x 2 + y*+z*) 2 



The attraction of the primary on matter at the same point has 



M£ 2 



the value ™ — ^ «- — 5- — 5- ; and the components of the 



D 2 ~ 2Dx + x 2 +y z + z 2 ' r 



disturbing force it occasions will be 



2Mk 2 x Mh*y Mfe 



J)3 ' J) 3 ' J) 3 9 ' * ' 



the squares and higher powers of x } y } and z being omitted. If 

 t denote the time, and v the velocity of rotation at the given 

 point, the centrifugal force will be 



^ or^VZI? 



As the rotation is supposed to take place in the same time as the 

 orbital revolution, Z 2 will (from the doctrine of central forces) be 



47T 2 D 3 



equal to , 2M , and accordingly the centrifugal force will be 

 expressed by 



while its components in the direction of the three axes will be 



If X, Y, and Z represent the sum of the components parallel to 

 each axis, 



„ mk*x 3M* 9 a? 



(^ 2 + 2/ 2 + ^) f "° 3 



y_ mk*y 



[afi + yi + z* 





(4) 



