740 Lord Kelvin on 



half periods of the first member of (78) and a single term of 

 its second membei . If e is < 1 by an infinitely small dif- 

 ference this approximation is infinitely nearly perfect. It is 

 so nearly perfect for e='9 that fig. 12 cannot show any 



'4 S *t> *7 «8 *9 Qj 



I ' r" | ■ 1 1 



Fig. 12; e = -9. 



deviation from it, on a scale of ordinates 1/10 of that of 

 fig. 11. The tendency to agreement between the first member 

 of (78) and a single term of its second member with values 

 of e approaching to 1, is well shown by the following modifi- 

 cation of the last member of (74) : — 



^l-^cos^ + ^ 2 ~^(l-^) 2 + 4^sin 2 i(9 ' ^ Uj * 



Thus we see that if e=\, II is very great when 6 is very 

 small ; and II is very small unless 6 is very small (or very 

 nearly =2i7r). Thus when e=l, we have 



1 j(l-. 2 ) 

 gc * (l-*) 2 -f*0 2 ^ 1)s 



which means expressing II approximately by a single term 

 of the second member of (78). 



§ 44. Return to our dynamical solution (75) ; and remark 

 that if J is an integer, one term of (75) is infinite, of which 

 the dynamical meaning is clear in (70). Hence to have 

 every term of (75) finite we must have J=/-|-S, where j is 

 an integer and 3 is < 1 ; and we may conveniently write (75) 

 as follows ; — 



