106 
PROFESSOR DONKIN ON THE 
become simply a'= [Cl, «,•], &(=[n, &,]. 
In these expressions Cl, a i7 b t are supposed to be expressed in terms of the variables. 
Now 
[n, «,]= 
L J J d{y jt Xj) 
but, by equation (26.), art. 10, this is equivalent to 
at) 
j d ( bj , fly)’ 
in which fl is expressed as a function of the elements and t ; and this last expression 
d£l 
obviously reduces itself to the single term — — ■ In like manner the expression for 
[H, £,■] reduces itself to ; thus the equations for determining the variation of 
the elements are 
jrs jn 
(E.) 
/ d£h -j > d£h 
Qi ~~~w i ’ Iki* 
in which Cl is to be expressed as a function of the elements and t*. This will be a 
sufficient account of the method for our immediate purpose. 
39. The following propositions in spherical trigonometry will be required. If a, b, 
c be the sides, and a, 0, y the opposite angles of any spherical triangle, then 
cos ( a-j-b ) 
(cos a. + cos /3) 2 
1 — cos 7 
■ 1 — cos « cos /3 
cos ( a—b) = 
sin « sin /3 
(COS a — COS /3) 2 
— V ; — + 1 — COS « COS B 
1 + cos Y 
sin a sin /3 
and if the sides be considered as functions of the angles, then 
. . (40.) 
• • (41.) 
da db , „ n dc , . n N 
-=cosy^+oo^ T? (42.) 
^=cosy^+cos(3^ (43.) 
dry dy dy 
The two last are easily verified ; but as the others are not so obvious, I shall give the 
demonstration. Putting x for the expression on the right of the equation (40.), we 
* The history of these remarkable formulae may, I believe, be stated as follows. They were first discovered 
by Lagrange in the case in which a,, bs were the initial values of x» yi, and £2 contained x x , &c. but not y,, &c. 
They were extended by Sir W. R. Hamilton to the case in which H contains both sets of variables; and 
finally, by Jacobi, to the case in which a lt &c., b lt &c. are any system of conjugate elements. Jacobi how- 
ever does not appear to have published a demonstration of them, and the only one which I have seen is by 
M. Desboves, Liouville’s Journal, vol. xiii. p. 397, and differs essentially from that given in the text. 
Sir W. R. Hamilton has pointed out the circumstance, that when £2 contains both sets of variables, the vary- 
ing elements determined by the formula (E.) are not osculating. 
