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PROFESSOR H. S. HELE SHAW OR THE THEORY OF 
in theory, effected appears to be that commonly referred to as the “ disk and roller,” 
with its modification, the disk-globe and cylinder-mechanism, or apparatus such as 
cones and belting which when analysed are all found to depend for their action upon 
the same kinematic principles. The action of the mechanism is simple enough, the 
disk, which is usually the piece to which motion is primarily imparted, drives the 
roller with a speed proportional both to its own angular velocity, and to the distance 
of the roller from its centre. Pieces of apparatus, such as cones and belting, apart 
from their unavoidable inaccuracy, are only suitable for the latter of the two above 
operations, viz., integration, and will therefore not be further alluded to. 
The disk and roller mechanism is chiefly known as a mechanical integrator. Its 
application for this purpose is due to Poncelet, who first suggested its use for ergo- 
metrical purposes as set forth in his ‘Mecanique Industrielle,’ and it was thus employed 
by General Morin. In this case it was used to perform the operation JF ds, where F was 
the force of traction measured by a spring, ds being the increment of space passed over 
at that instant by the cart. It was soon afterwards employed by Professor Moseley 
in a “ constant steam engine indicator ” designed for the use of a committee of the 
British Association.* In this instrument the disk was replaced by a cone, and F was 
the varying pressure of steam in the cylinder. Since then the disk and roller 
principle has been adopted by many inventors for purposes of integration in con¬ 
junction with dynamometers for measuring the power transmitted from a prime mover 
or to a machine. The cone and roller principle was employed by Professor Moseley 
in a calculating machine,! and by Sang and others as a Platometer or Planimeter. 
Messrs. Ashton and Story have revived the application of the disk and roller to a 
continuous Indicator, which should, however, really be called a steam-power Integrator . 
In the last-named instrument advantage is taken of the fact that as the steam 
pressure acts alternately on one side or the other of the piston, the roller may be 
made to take a corresponding position on either side of the centre of the disk, thus 
both the forward and backward stroke of the engine combine to effect the continuous 
rotation of the roller in one direction, and thereby integrate the total work done. 
All the foregoing examples are applications of the disk and roller mechanism for 
purposes of integration. Differentiation may practically be effected by means of the 
disk and roller in the following manner:—In fig. 1 let G represent the disk, and B 
the movable roller. Instead of the roller B being fixed to its axis the latter is a 
screw (E E) of which B forms the nut. For convenience the disk is supposed to be 
driven from the hollow shaft F by means of the roller A, which is always at a 
constant distance from the centre of the disk, and may be therefore alluded to as 
having the same motion as the disk. Suppose this screw (EE) to be turned by one 
body whose rotary motion varies with the rate of increase of one of the two variable 
quantities, while the motion of the disk depends on the rate of increase of the other 
* lltli Report of the British Association, p. 308. 
f L. E. and D. Phil. Mag., vol. xxx., p. 171. 
