396 
PROFESSOR H. S. HELE SHAW ON THE THEORY OF 
Let 
Then approximately 
n = number of revolutions of RQ 
l = pitch of its screw 
Volume of solid — nl\vd:. 
By making l very small the result may be made to approach the true value as 
nearly as desired. 
The operation 
| y)dydx 
may, however (in theory), be performed in the following way. 
Fig. 29. 
Let A C B, fig. 29, be as before the section of any solid. Then to find its volume 
the foregoing expression would (if possible) first be integrated with respect to y, and an 
expression of the form Volume= j\jj(x)dx obtained. The expression i fj(x) may be called 
the ordinate of a curve of areas which may be represented by A D B E, the ordinate 
D E —y—fx) at any point of which gives the numerical value of the area for that 
value of the abscissa x. By passing the pointer of the simple integrator over the 
curve ADBE, the volume of the solid is given at once. 
To apply this reasoning to effect a mechanical solution, the pointer of a simple 
integrator must be passed with sufficient rapidity round the solid as it moves along 
dA 
the axis of x, so that the differential coefficient - = average areas can be obtained, 
dt & 
this gives at any instant the ordinate (D E), which can be used as above stated by 
means of a suitable mechanism. 
