490 Mr. D. Vaughan on Static and Dynamic Stability 



a time different from that of its revolution, would have its equa- 

 torial gravity subject to a variation of about If per cent.; 

 and the central pressure along the plane of the equator would 

 undergo a periodical change of about 3000 pounds to the square 

 inch. Now in consequence of the compressibility which belongs 

 to every kind of solid matter, the satellite would be continually 

 changed to an ellipsoid, the longest diameter always forming the 

 same angle with the direction of the primary. If its component 

 parts had a modulus of elasticity as great as that of iron, a dif- 

 ference of one-fifth of a mile may be expected between the major 

 and mean axis ; but were the mass of a more yielding character, 

 or were it covered with fluid, its ever-changing form would de- 

 viate more considerably from a true sphere. 



The effect which the attraction of the primary would exert oil 

 the rotation of an ellipsoidal satellite, the major axis of which 

 had a constant inclination to the radius vector of its orbit, may 

 be found by a method similar to that pursued for determining 

 theoretically the amount of the precession of the equinoxes. Let 

 A, B, and C be the major, mean, and minor semiaxes of the 

 ellipsoid, the last being perpendicular to the plane of the orbit, 

 and the first forming the angle -v/r with the direction of the pri- 

 mary. Supposing the satellite homogeneous, the change in the 

 velocity of rotation at the extremity of the major axis in a unit 

 of time will be expressed by 



f^(A-B)sin2^ ..... (1) 



D being the distance of the primary, and M the measure of its 

 attractive power. According to the theory of central forces, 



M 4«7T 2 



1^3 = 7p2~, 7r being put for 3-1416, and T for the time of revo- 

 lution ; the expression for the change in the equatorial movement 

 thus becomes 



^2 -sm2f (2) 



Now, for a synchronism of the orbital and diurnal motions, the 

 equator must have a velocity equ 

 the last expression, there results 



equator must have a velocity equal to -^ ; and dividing this by 



TA 



T,== 37r(A— B)sm2^' • • • • ( 3 ) 



T' denoting the time in which a satellite, having no primitive 



rotation, would acquire one sufficiently rapid for keeping the 



