558 8ir W. Thomson on Periodic Motion 



have been hitherto considering. It is accordingly convenient 

 now to modify our notation as follows : — 



*'=** x'=K, **=*; r=6, y'=Vi, £'=S ) (9)i 



Here (£ z -, ^, ^ are the generalized components of distance 

 from 0, at the ith transit through ty = 0, of the system 

 pursuing its path infinitely near to the orbit ; and f ., 97. , ?. 

 are the corresponding momentum-components. With the 

 notation of (9), equations (8) become equations by which the 

 values of these components for the ^ + lth time of transit 

 through tJt = can be found from their values for the ith 

 time. They are equations of finite differences, and are to be 

 treated secundum artem, as follows : — 



29. Assume 



<!>i+i=p<l>i, x i+ i=PXi> $ i+ i-p$i 

 &+i=p6i y i+ i=pvi> S i+1 =pSi 



Substituting accordingly in (8) modified by (9), and elimi- 

 nating f ., 7]., 5., we find 



? i; j. • • • (10) 



o>. 



/ 24 \ / 25 \ / 26 \ 



(21+-+51 j o + 54J9+f22+- + 52 j0 + 55l x + (23 + --+53p+56)^== 



r31 + - + 61p + 64\ + (32+- + 62 jO + 65) x +(33+- + 63p + 66^=0j 



Kemarking that 41 = 14, 12 = 21, &c, we see that the deter- 

 minant for the elimination of the ratios <j)\x\$ is symmetrical 

 with reference to p and l|p. Hence it is 



C3(p 3 + p- 3 ) + C 2 ^ + P - 2 ) + C 1 (^ + ^) + 2C , . (12) 



where C , G,, C 2 , C 3 are coefficients of which the values in 

 terms of 11, 12, &c. are easily written out. This determinant 

 equated to zero gives an equation of the 6th degree for 

 determining p, of which for each root there is another equal 

 to its reciprocal. We reduce it to an equation of the third 

 degree by putting 



p + p- l = 2e (13) 



Let <?!, e 2 , e 3 be the roots of the equation thus found. The 

 corresponding values of p are 



*i± Vfe 2 -1) ; e 2 ±i/(e 2 2 -l) ; %±s/fe 2 -l). . (14) 



