31*2 
PROFESSOR DONKIN ON THE 
(50. The solution of the system (A.) of the last article will now be expressed by the 
same equations (i.), (ii.), if the elements h, c, r, nx be variables defined by the system 
(see art. 58.) 
h'=- 
dW 
dr ’ 
d_W 
d'ST ’ 
dW 
dh 
j 
dW 
dc ’ 
where W is to be obtained by substituting in the expression (W.), art. 59, the 
values of g', 0, u, v in terms of the elements and t, derived from equations (i.), (ii.). 
1’he I'esult of this substitution is 
W=-4-- 
n ^ A 
1 
2 ' 4/2 4w2/2 2 
\/ w^c^.cos '2n{t-\-r) 
+ — ^^^^-cos 4w(/+r)-|-w^/(/— \//“ — f), 
in which the value of f in the last term must be understood to be substituted from 
the first of equations (ii.). If we call <p the angle between the pendulum and the 
vertical, we shall have evidently 
'rfl{l—s/ — — CO’S, (p), 
and the differential coefficient of this term with respect to any constant h involved 
d{q^) 
2 cos ® dk 
in the value of § null be 
Observing this, we obtain the following expres- 
sions for the variations of the elements : 
/i'= — n(sec <p — 1 )\/ -w^c^.sin 2 n{t-\-T)- 
n/2 
sin An{t-\-r) 
h 
2 
t-l(secp-l)(H 
h 
V h^—n^c^ 
- cos 2n{t-\rr ) ) cos iri{t-\-7) 
c' = 0 
-o'i2-J(sec?5-l) 
2«(^+r) cos 4/2(/+r). 
The third of these equations gives ab — constant ; hence, by nieans of the equations 
at the end of art. 59, the following expressions are easily deduced : 
y 
— (secip— 1) sin2n(/-l-r)4 
T =- 
4/2 
sin An{t-\-T) 
+ 
( - 1 -b cos 4w(^ -b^)) +' 2 (sec oos 2 « (^ +t) j 
, , fl— cos4n(/ + T) , ,cos2n(/ + T)] 
=naH ') r 
These equations are rigorous, and in general not easier to integrate than the original 
system of which they are a transformation ; but they may be integrated approxi- 
