673 
of Edinburgh, Session 1871-72. 
conclude that k, k,... are constants. [They are sometimes called 
the “cyclic constants (Y. M. §§ 62 — 64)]. The equations of motion 
(15) thus become simply 
dip J1Q 
dt df 
+ 0 
f / da dy\ /da' d(3'\ ) 
{ K y(£0 dif/J + K \dO dif/J^ ) 
+ & c. 
with corresponding equations for rj 0 , 4, and with the following 
relations from (7), between to, y 0 -" and if ^ 
7. Let 
dQ . dQ ^ 
dt o drj 0 ~ 
dQ 
dt o" 
0 , &c. 
• (17). 
'da d/3\ / da d/3'\ 
dp~*j') + K \d? ~T^) h &G -> be denoted {?,</>} ■ 
(18), 
so that we have 
• - • (19)- 
These quantities {<p, if/} , {0, if/} , &c., linear functions of the cyclic 
constants, with coefficients depending on tbe configuration of the 
system, are to he generally regarded simply as given functions of 
the co-ordinates if >, <p, 0, ... : and the equations of motion are 
3F + 3? + ta + 
( 20 ). 
In these (being of the Hamiltonian form) Q is regarded as a 
quadratic function of to, rj 0 , £ 0 -** with its coefficients functions of 
i ft, <p, 0, &c. ; and applied to it indicates variations of these co- 
efficients. If now we eliminate to, Vo, to’" from Q by the linear 
equations, of which (17) is an abbreviated expression, and so 
have Q expressed as a quadratic function of ij/, <p, 0,.- , with 
its coefficients functions of if/, <p, 0, &c. ; and if we denote by 
dQ dQ 
dp’ dif/ ’ 
&c., variations of Q depending on variations of these co- 
