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VI. The Theory of Continuous Calculating Machines and of a Mechanism of this 
class on a New Principle. 
By Professor H. S. Hele Shaw. 
Communicated by Professor Sir William Thomson, F.R.S. 
Received and Read June 19, 1884. 
There are an almost unlimited number of applications for a mechanism by means of 
which the velocity ratio between two rotating pieces could either be determined at any 
instant, or be made to vary in any required manner. Such a mechanism would enable 
two variable quantities to be dealt with numerically, for with it the operations of 
differentiation and integration could be mechanically performed. 
For differentiation it would be necessary to cause two rotating bodies, which might 
be disks or rollers, to be turned at speeds which varied respectively according to the 
rates of change of the variable quantities, one quantity being a function of the other, 
when the velocity ratio, shown by a suitable index, would give their differential 
coefficient. This is evident, because velocity is simply the space passed over in a 
given time, and is, in the limit, proportional to the increment of this space. For 
integration, while one body is moved at a rate which changes with the independent 
variable, the velocity ratio of the two bodies (i.e., the two rotating disks or rollers) 
must be made proportional to the dependent variable. The actual velocity of the 
second or driven rotating body then becomes a measure of the product of the latter 
into the rate of change of the former at the same instant. The motion of the driven 
body, as recorded for any period of time by a suitable index, is therefore a measure of 
the integration of their product for that time. 
The primary or simple form of mechanism has but small value in comparison with 
the possibilities which a combination of such mechanisms seems to offer. The index, 
which shows the differential coefficient, or the driven disk which records the results of 
integration, can have respectively but one position or rate of motion at any instant. 
By employing, however, a suitable series or chain of such mechanisms, if need be of 
both kinds, the final index or the final disk could be made to either indicate or record, 
as the case might be, the result of any required number of conditions. It is difficult 
to say what limit there would be to the powers of the continuous calculator which 
such a combination would form. 
The only hitherto known mechanism by which both the foregoing operations can be, 
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