1891.] On Instability of Periodic Motion. 197 



These equations allow us to determine the three displacements, 0, %, #, 

 and the three corresponding momentums, , ?/, *, for any position on 

 the path, in terms of the initial values, 0', x', #', ', ?/', " supposed 

 known. 



6. To introduce now our supposition (2) that P'oP is part of a 

 periodic path ; let Q be a position on it between P' and P ; and let 

 us now, to avoid ambiguity, call it P'QoP- Let P' and P now be 

 taken to coincide in a position which we snail call ; in other words, 

 let P'QoP, or OQO, be the complete periodic circuit, or orbit as we 

 may call it. Our path. P'P is now a path infinitely near to this orbit, 

 and P' and P are two consecutive positions in it for which ^ has 

 the value zero. These two positions are infinitely near to one another 

 and to 0. We shall call them O;, and 0;+i, considering them as the 

 positions on our path in which ^ is zero for the zth time and for the 

 (*-f l)th time, from an earlier initial epoch than first passage through 

 YT = o which we have been hitherto considering. It is accordingly 

 convenient now to modify our notation as follows : 



=fc 1 



f 



= +1 J 



( y ) 



X = 



Here 0i, %;> -^t are the generalised components of distance from 0, 

 at the ifch transit through fy = of the system pursuing its path 

 infinitely near to the orbit; and , 1;;, ^ are the corresponding 

 momentum components. With the notation of (9), equations (8) 

 become equations by which the values of these components for the 

 i-\- 1th time of transit through ty = 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 : 



7. Assume 



0i+l = p0i, X*+l = PXh &i+l f>$i J 1 



ft+i = p, tji+i = pt]i, +1 = /> J 



Substituting accordingly in (8) modified by (9), and eliminating 

 & % , we find 



^ = 



34 . _ . _.\ , . / . 35 . \ . / .36.. 



= j 



Remarking that 41 = 14, 12 = 21, &c., we see that the determinant 



(11). 



VOL. L. 



p 



