100 
PROFESSOR DONKIN ON THE 
It is obvious that we may change the signs of the first pair. And generally, that 
if f, g he any two conjugate elements, we may substitute for them kf, where X is 
any determinate constant, i. e. not a function of the elements*. 
The above elements coincide with those given by Jacobi. My object has been 
merely to illustrate a mode of obtaining them which seems capable of useful applica- 
tions. 
31. As a second example I shall apply the method to the case of the motion of a 
solid body about a fixed point. 
Let the fixed point be taken for the origin, and the principal axes of the body 
through that point for the axes of x, y, z. Let |, q, ^ refer to the same origin and to 
axes fixed in space ; a, b, c being the direction-cosines of the axis of x referred to 
the fixed axes of |, ??, and a', V , c ' ; a", b", c" being respectively the direction-cosines 
of the axes of y and z. Let 0 be the inclination of the plane of x, y (or “equator”) 
to that of ri (or “ ecliptic”) ; the longitude of the node, reckoned from the axis of 
and <p the right-ascension of the axis of x. Then if A, B, C be the Moments of 
Inertia, and p, g, r the angular velocities, about the axes of x, y , z, the expression for 
the vis viva is T=^(Ajt> 2 +B9 2 +Cr 2 ), where 
p — — 0' cos <p — sin <p sin 0 
q—ti sin <p—dg' cos p sin 0 
r=p>'-j-\|/ cos Q- 
Let u, v, iv be the variables conjugate respectively to 0, <p, \p, so that 
dT dT dT 
U — - 1 7 T 1 V — —.1 W=~ n , 
dtf d<d dty 
the following expressions will be found without difficulty 
a , sm <p . . 
A p——U COS <p -f- • ~r (V COS 0 — IV) 
sin 6 
■n • . cos® . 
liq = u sin (v cos 0—w) 
C r=v. 
Considering at present only the case in which no forces act, we have the integral of 
vis viva T =h, which becomes 
COS p + ^ (V COS 0-w) ) 
1 / . .cos <p \ 2 
+ g sin <p + ^ 0 cos 0 — «>) ) 
+d'= 21 ' (■•) 
* Alore generally, we may substitute for /, g any two functions of them,/), q, such that 
dp dq dp dq _ 
dg df df dg 
a condition which requires the solution of a linear partial differential equation for the determination of one 
function, if the other be assumed. But on the subject of the transformation of elements see below, arts. 34, 35. 
