220 



cases there will be other conditions, which will be seen thus : — Let 

 N(p-q) represent the value which No takes when the relation between 

 n and n l is such that^m = qn l . Then, when this is the case, the equa- 

 tion for o> 3 becomes 



= ffi(p-q) + -P sin i* q sin (pe) + 0 sin t* q cos (pe) + periodic terms. 



'Now, the terms sin (pe) not containing the time, are constant quan- 



tities, and to find, when — is 0, we mast equate the three terms given 



above to zero, and this will give us an equation for determining what e 

 must be in order that the body may revolve permanently with the relation 

 pn - qn 1 . That such a condition maybe possible, will, of course, imply 

 that the coefficients, &c, in the above equation, have such a value as 

 to give values of sin pe and cos pe not greater than unity. And it is 

 evident, in order that such a requisition may be fulfilled, the coefficient 

 of either of them cannot be small with respect to JV— . And this will 

 show, that though there are several such relations, there can only be 

 a limited number ; for, as the quantities p and q become large, so also 



does the power of — or — -, which multiplies such terms, become 



large; and hence for large values of^? and q, the coefficients in ques- 

 tion rapidly diminish, the more so when i is small, and there will not 

 be many of them which are larger than JV pn . qn , and consequently not 

 many different conditions of stable rotation. Mow many there are it is, of 

 course, impossible to say without more knowledge than we have, or ever 

 can have, of the numerical values of the various quantities concerned. 



We may conclude, then, certainly, that there will always be either 

 an acceleration or retardation of a body revolving freely about a fixed 

 point, and acted on by a^disturbing force moving round it, except when 

 certain given relations exist between n and n 1 ; but which of the two it 

 will be it is quite impossible to say, without knowing more about the 

 form of the revolving body than we do of the moon ; for without such 

 knowledge we cannot determine the algebraic signs of the various co- 

 efficients. One thing, however, appears highly probable, and it is this: 

 that if the conditions of equable rotation can be satisfied by values of 

 n not very much greater than n 1 such as n = fra 1 , or, on the other hand, 

 by values not very much less than n 1 , such as n = fn 1 , if its rotation had 

 ever been very much greater, or very much smaller than it is, it would 

 seem that the change ought to have ceased when it came to a position 

 of equable rotation, such as either of the former, without further 

 diminishing or increasing till it became equal to n l ; and from hence 

 it would appear that its rotation can never have been very different 

 from what we actually observe it to be. 



