218 



proximate values, including those arising from perturbation, have been 

 substituted for i, <j>, and 0 l in the general equation. When this has been 

 done, its general form will be 



—2 = iV 0 + A x cos (n - n 1 1 + e) + B x sin (n-n l t + e) 



+ A 2 cos (2w -2nt + 2e) + B 2 (sin 2n-2nt + 2e) +, &c 

 + P sin i p ~ 9 cos(p-qt + qe) + 0 sin t L p q sin - q t - qe) +, &c. 



where N Q is the constant part, and where A lf &c., do not contain sin t 2 

 as a multiplier, and where the last line represents the general form of 

 those terms which are multiplied by sin i l} of which the terms in the 

 expansion of N which have been used are an example, viz. 



„ 15 



— sin i — cos (2n - n l t + i), &c. 



r 3 2 



Let us omit for a moment the periodic terms, and consider the value 

 of Nq, or rather that part of it found above, and which will be the only 

 part when sin «, = 0. This term may be put under the form 



S 1 /1 ] 1 \ 



How this changes its sign, first, when the latter factor does, i. e. when 

 I2n- ln x becomes 0 ; that is, when n is something less than |ti\ This, 

 therefore, would, as far as this term is concerned, be one condition 

 under which the rotation would be stable ; but this would be no more 

 than an approximate value, because it does not take account of terms 

 multiplied by sin 2 i, which, for the moon, though small, is not 0. 

 Again, it changes its sign when n - n 1 does; that is, supposing the co- 

 efficient of to be positive, it would cause an acceleration or retar- 



n - iv 



dation according as n was greater or less than n 1 ; but in order that there 



might be stable rotation when n = n l , it is evident that — 2 ought to 



change its sign by passing through zero, not by passing through infi- 

 nity, as it appears at first sight to do when n - n x = 0. Let us examine 

 what the true value is under such circumstances. It is quite evident 

 that it must be either 0 or infinite, since it is only by passing through 

 one or other of these that it can change its sign. To see which it really 

 is, it will be necessary to look back to the process by which the func- 

 tion containing it was formed ; and if we do so, we shall see that the 

 term is in reality only the first of a series, consisting of odd numbers of 

 n - n 1 , having its signs alternately positive and negative ; so that 



