376 
PROFESSOR H. S. HELE SHAW OR THE THEORY OF 
Let 
y— 0 then co = 0 
y=z=R—y „ &>=aq 
y =H ,, oj= cc ( i . e., may approach that value). 
Though the greatest possible range is thus obtained the result is not really so 
convenient for practical purposes. For instance, with an ergometer or steam engine 
integrator one advantage of the former mechanism is lost, for now work is not 
registered in simple proportion to the deflection of the spring. For the converse 
purposes the scale would merely have to be graduated according to the equation 
betweed co and y (oq being constant). 
There is, however, a connexion between the two forms of disk and roller which 
bears a resemblance of considerable interest to two corresponding forms of sphere and 
roller, and when followed out leads to important results. 
In fig. 6 let 0 be the centre of a sphere which rotates on two fixed centres 
C and Cq that is about the axis C Cj. 
Let a be the point of contact of the roller A which works against, or rolls upon, 
the great circle of revolution (i.e., the equator), and b the point of contact of the roller 
B which rolls on a small circle whose radius is bq. 
Let angle of plane of rotation of B with axis CCpa 
then 
but 
therefore 
angular velocity of B _ w 
angular velocity of A &q 
bq 
0 a 
bq 
— = — = sm a 
\Ju 
bq 
0 a 
to 
— sm a 
or 
(o=aj 1 sin a 
In fig. 7 the roller A' is movable as well as roller B'. They are both attached 
to the same movable frame—their planes of rotation being always perpendicular to 
each other. 
Then 
angular velocity of P/ o' b'q' 
angular velocity of A' co l a'p' 
but by similar triangles O'q'b', 0 'pa 
t / r\' / 
a p —\Jq 
