386 
PROFESSOR H. S. HELE SHAW ON THE THEORY OF 
directly obtained in the following way. Suppose the rollers A and B (fig. 22) to he 
graduated and a pointer or index attached to each so as to enable the distance turned 
through by either to be read. Suppose each reading to be brought to zero and the 
pointer P (connected with the movable frame) moved into a position such that 
AP = x= one factor. 
Then turn the roller A through a distance such that the index on A reads 
m — the other factor. 
Then by taking suitable units, or what amounts to the same thing, by using 
suitably proportioned rollers, 
Reading of B — n = mx = product of the two given factors. 
To obtain a quotient a similar method would be adopted, but the roller B would 
now be turned instead of A, then 
Reading of A = m =-= quotient. 
cc 
The practical application which suggests itself is that of the reduction of tables, in 
which case one factor is constant. The work might be rapidly performed by always 
turning A or B (as the case might be) through a constant distance from zero, the 
pointer P having been first set to the other factor, to an adjustable stop when the 
reading of the driven roller gives at once the required result. 
Thus far the working is only a special case of that of the integrator, viz., where the 
area is a rectangle, but with the form of instrument for the converse process, that is 
with the roller A screwed upon its axis, a new principle can be brought into operation. 
Suppose the roller B (fig. 22) to be turned through a distance n, then the roller A 
will in consequence turn through a certain distance m, but inasmuch as it forms a nut 
upon the screw A P, it will at the same time continually alter the position of the pointer 
P, and consequently the value of x, thereby changing the position of the movable 
centres Tck, or axis of rotation of the sphere. Thus the reading (to) of the index of A 
is a value which is no longer a simple product or quotient but of a nature which must 
be investigated. 
Let 0 e 6 l be the angles turned through by A and B respectively. 
Let 
l = pitch of the screw AP. 
a == radius of roller A 
b = „ „ B. 
x = distance AP. 
k = „ AO. 
b(W l 
add 
tan a =7 
k 
Then 
