and 
we have 
DIFFERENTIAL EQUATIONS OF DYNAMICS, ETC. 
105 
^(yrfx^—dX (by (5.)), 
2,(^0 —d(— X+S^a?#,-)), 
an equation which must become identical when x„ .r 2 , &c. on each side are expressed 
in terms of y x , y 2 , &c. But the right side being then a complete differential of a 
dxj 
= -r" must 
ayi 
subsist. The investigation of art. 2 shows that they do subsist, and is therefore 
perhaps to be preferred. 
function of y } , y 2 , &c., the left side must be so also ; hence the conditions 
dxi 
dyj 
Section III . — On the Equations of Rotatory Motion. 
3 7. In this supplementary section I propose further to exemplify the preceding 
theory by exhibiting the application of it to the problem of rotation in a more detailed 
form than was consistent with the plan of the former part of this essay. For this 
purpose it will first be desirable to anticipate the subject of a future section, so far as 
to give a concise deduction of the method of the variation of elements in its simplest 
form. 
38. Variation of Elements . — Suppose a complete normal solution of the system of 
differential equations 
/ dTi / | A 
*■=«’ * + sr 0 
(i.) 
has been obtained, so that we have '2n elements, divided into two conjugate sets 
a x , a 2 , ... a n \ b x , b . 2 , ... b n 
as in the former articles, so that 
O, 6i] = 1 , O, «,] = [h bj\ = [a t , bj\ =0. 
It is required to express the solution of the system 
x 
dTi | d£l 
dyi dyl 
/ . dTi , dQ r. 
y< + s+s , =0 
(I.a) 
in the same form by means of variable elements. The disturbing function H may be 
a function of all the variables x„ &c., y 1} &c., and may also contain t explicitly. 
In the undisturbed problem we have a-= 0, ^-=0 ; i. e. the equations 
§+(Z,«,]=0, §+[Z,i,]=0 (e.) 
(see art. 22.) subsist identically when x 19 &c., y„ &c. are expressed in terms of the 
elements and t. 
In the disturbed problem, x l} &c., y x , &c. are to be the same functions of the 
elements and t as before ; hence the equations (e.) continue to subsist identically, and 
therefore the values of «■, b\, namely, 
«i=^y+[Z, + a~\ 
4;=§+[z,i 1 ]+[n,i i ], 
MDCCCLIV. 
P 
