388 
PROFESSOR H. S. HERE SHAW ON THE THEORY OF 
Again, suppose P (fig. 23) to be taken on the other side of A. Then by similar 
reasoning since now 
held, , k 
. = tan a=- 
adO x 
dQ V - 
2nrak dd 
and by integration 
hi 6 
ft, = ^ log, ft 
If, as before, m and n are the respective readings at the end of the operation, and 
the pitch of the screw be such that 
27 rak 
Then 
^ = modulus of the Naperian system of logarithms. 
71 = log 10 m. 
In order to graduate the dials in the latter application, since 6 can never be zero the 
limiting position can never be reached ; but when 
m= 1 then n = 0. 
therefore the dials of A and B must be adjusted so that these two conditions are 
simultaneously fulfilled. 
In this way it is possible to find the logarithm of a number m to any base by merely 
turning the roller A through that distance and reading the dial of B. 
There is another mode of obtaining any root, which will be easily understood 
when it is remembered that by using n sets of spheres and rollers, the value of 
(x 1 X 3Co X Xo X . . . x,) can be obtained. Make these values of x all equal; then with n 
frames x n is given. By turning the last wheel through af, and keeping the frames in 
suitable positions, the first wheel of the series will turn through a distance x. 
Thus, if the last wheel be turned through a distance N. 
Beading of first wheel or roller 
= m = \/N. 
The application of a mechanism to obtain the foregoing results had been previously 
suggested by Professor Moseley in a paper already alluded to (p. 368), but the foregoing 
investigation was made without any knowledge of this. The modification of Professor 
James Thomson’s disk- globe- and cylinder-mechanism (figs. 3 and 4) may be at once 
applied to obtain similar results. 
Applications for continuous calculation. 
The foregoing applications are for the purposes of obtaining numerical results in 
which the mechanism is employed, rather as a discontinuous than a continuous 
