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PROFESSOR H. S. HELE SHAW OX THE THEORY OF 
j*F 1 (a:)F a (a;) .... F n {x)dx 
where n is the number of sphere and roller mechanisms, each frame being made to 
assume a position depending on the respective function with which it deals. 
The volumes of solids of revolution can be obtained (with two sets) by merely 
passing the roller A in direction of the axis of x, and keeping the pointer P on 
the surface, so that at all times 
y— radius of circle of revolution. 
Then 
volume= 7 t y~dx 
and 
Also with ruled surfaces the pointer of 1st set is kept on one surface, so that 
?/ ( = one ordinate = y=< 5 k(:r) 
y»= the other =z=\jj(x) 
Reading of index of C = 
= volume — ((/> (x)\jj(x)dx 
Again, in the integration of trigonometrical functions, which are of great importance 
in naval architecture, it is only necessary to keep the pointer of the index w r hich 
is attached to the movable frame, and so controls the axis of rotation at the angle, the 
trigonometrical ratio of which is a function of x, as for instance, in the equation 
y = | tan a dx 
which gives the area of a curve in which 
cl— angle of heel of vessel 
x— corresponding ordinate. 
All the ratios could be in this way easily dealt with. 
The converse application of the sphere and roller mechanism will easily be under¬ 
stood, as the principle has already been fully dealt with in the case of the disk 
and roller, though only the one example of the speed indicator has yet been 
ds 
suggested—in this case the position of the pointer indicates v= 
Suppose, however, that a single sphere and roller arranged for the purpose be 
applied to a single integrating mechanism which is giving the work done in a steam 
