ON THE STABILITY OF THE MOTION OF SATUKN's RINGS. 307 



Substituting these values in equation (18) and dividing by Jfa 4 /", we obtain 

 (!_/.) w< + (l_|/ + i /^) wV + (|-6/'-^-i7r + 2/ 2 5 r)^ = ...... (28). 



The condition of stability is that this equation shall give both values of n 2 

 negative, and this renders it necessary that all the coefficients should have the 

 same sign, and that the square of the second should exceed four times the 

 product of the first and third. 



(1) Now if we suppose the ring to be uniform, f, g and h disappear, 

 and the equation becomes 



7i 4 + ttV + | = ................................. (29), 



which gives impossible values to n 1 and indicates the instability of a uniform 

 ring. / 



(2) If we make g and h = 0, we have the case of a ring thicker at one 

 side than the other, and varying in section according to the simple law of sines. 

 We must remember, however, that f must be less than \, in order that the 

 section of the ring at the thinnest part may be real. The equation becomes 



The condition that the third term should be positive gives 



/'<-375. 

 The condition that n' should be real gives 



71/ 4 -112/' + 32 negative, 

 which requires/' to be between '37445 and 1*2. 



The condition of stability is therefore that /" should lie between 



37445 and "375, 



but the construction of the ring on this principle requires that f" should be 

 less than '25, so that it is impossible to reconcile this form of the ring with 

 the conditions of stability. 



(3) Let us next take the case of a uniform ring, loaded with a heavy 

 particle at a point of its circumference. We have then g = 3f, h = 0, and the 

 equation becomes 



392 



