344 
Proceedings of the Royal Society of Edinburgh. [Sess. 
Combining these, we get 
x— — a(\j/ + <j> cos 0) sin <j b - a sin 6 cos cf> . 0 \ 
y = a(ij/ + <j> cos 0) cos cf) - a sin 0 sin . 0 f . . . (53), 
z = a cos 0.0 / 
which are the kineftiatical conditions. 
The kinetic energy of the motion of the centroid is 
T c = \ m(x 2 + y 2 + z 2 ) 
= sin 0 cos </> . 6 + cos 0 sin + sin </> . \j/) 2 + ( - sin 0 sin <£ . 6 
+ cos 0 cos <jf> . cj> + cos c j> . if/) 2 + cos 2 0 . # 2 ] (54). 
This reduces to 
T c = fyna 2 (6 2 + cos 2 0 . </> 2 + ^ 2 + 2 cos 0 . <jnj/) . . . (54'), 
but for our present purpose it is necessary to leave it in the expanded form. 
Along with this we have the kinetic energy of rotation 
T r = 4{ A(^ 2 + cj> 2 sin 2 0) + C(xf/ + cos 0) 2 } . . . (55); 
where C is the moment of inertia of the body about its axis of symmetry, 
and A that about any other axis at right angles to the axis of symmetry 
and passing through the centroid. Then 
T = T ( . + T r (56). 
27. It will be seen from an examination of the expression for T c above, 
that the integrability conditions are fulfilled as between 0 and cp, and 6 and \fs. 
As regards T r the relative co-ordinates are integral functions of 6, cp, \fr ; 
and the ordinary methods apply. Hence the 0-equation for the hoop or 
disk can be found in the ordinary way by the equation 
d 0T 0T 0V 
— — T - — . . . . . (57) : 
dtdO dO dO 
where V ( = mga sin 0) is the potential energy. 
The <p and \fr equations are, however, 
d 0T 0T r id d. ) 
Tj. ” ^7 + ma { & vX C0S # sm </>) - 2/^( cos o cos cf)) } =0 
at c<p ccf) { dt dt ) 
d 5T f • d/ • >\ , • d, , x I A 
dtU +ma \ x dt {sm ^ +y jf cos ^ } 
since dT r /d\Js — 0. The last two equations may be written out in full by 
the reader. The ^-equation reduces to 
d 0T o • a a I a 
r - ma - sm 6 . 6<p = 0. 
dt 0i p 
The 0-equation (57) worked out has the form 
(A + ma 2 )0 + (C + ma 2 - A)</> 2 sin 0 cos 0 + (C + wa 2 )^ sin 6 = - mga cos 0 (59). 
