219 



although, if we were to take only the first turn, it would certainly be 

 infinite when 71 - n l = 0 ; yet it does not follow that the entire series is 

 so. On the other hand, it is quite evident that it is not; for it arises 

 from the multiplication together of such terms as sin i and sin n. Now, 

 whatever be the value of c or x, sin i and x are never greater than unity, 

 and hence their product must consist of terms whose sum cannot be 

 greater than unity. In expanding sin ce, &c, we only took the terms 

 of lowest dimension that occurred, that is, we put 



i = i x + a, or sin i = sin (t x + a) 



sin i jl - {a 2 + | a 4 +| + COS f - |- - &c.^ 



The higher powers of the quantities represented by a were rejected ; if 

 they had been retained, we should have had a series such as that men- 

 tioned. If, then, the nonperiodic term cannot become infinite, and yet 

 changes its sign when n becomes equal to n l , it is plain that for such a 

 value it must disappear, and that as far as this term only is concerned, 

 n - n 1 = 0 will be approximately a condition of stable rotation. If, 

 however, we had taken into account terms depending upon the argu- 

 ments 0 1 20 - 0\ &c, it will be found that we should have introduced 

 into the constant terms quantities multiplied by sin 2 <, and which do 

 not change their sign when n = n 1 ; so that the entire value of iY 0 will 

 be a quantity which does not change its sign, and cannot become 0 when 

 n = n l ; and the relation between n and n 1 thus obtained by equating 

 JV 0 to 0 will be the relation which ought to be used instead of n = n x ; 

 but it would appear also that in addition to this relation, the conditions 

 will also be very approximately satisfied by n = n\ To show this, let 

 JV n=n i be the value which JST 0 assumes when n = n l . 



dw 3 



Then, the equation for — becomes 



^ = JV n=n i + A x cos e + B x sin e + A 2 cos 2e + B 2 sin 2e. 

 at 



A 2 may be rejected. Now, if we put this = 0 it will give us a value by 



which e may be determined so as to satisfy the equation — — 3 = 0 when 



n =?i x ; for, for no value of e can sin e and cos e be simultaneously equal 

 to 0 ; and since N n=n \ is very small compared with A x B x , and espe- 

 cially B 2 , it is evident that a possible value of e may be found to satisfy 

 the equation 



N n=n + A } COS e + &c. = 0. 



In other words, it must have a principal axis inclined at a particular 

 angle to the radius vector of the disturbing body ; and this angle, 

 though it appears to be small, cannot be 0. And these appear to be 

 the only conditions when the mean value of i is nothing ; but in other 



