278 Proceedings of Royal Society of -Edinburgh. [sess. 
and even the roughly constructed instrument shown will give 
results sufficiently accurate for most purposes. 
The construction of the differentiator depends on the w 7 ell-known 
fact that if the values of a variable quantity he represented on a 
diagram by the ordinates of a curve, its rate of change, at any 
point of the curve, is measured by the slope of the tangent at 
that point. 
The machine, then, is guided by hand so that one line on it 
remains tangent to the curve, while a tracing point describes on a 
second sheet of paper a curve whose ordinates are proportional to 
the slope of the tangent. Thus if y = f(x) be the equation to the 
original curve, the derived curve will have for ordinates the 
corresponding values of d(f(x))/dx. The abscissae are the same 
on both curves. 
In order that a line may be tangent to a curve it is necessary 
that two consecutive points on each should coincide. In practice, 
two black dots on a piece of transparent celluloid are used, the 
distance between them being about 2 mm. 
The plan of the machine is shown in fig. 1. It consists of 
three parts. Firstly, the large drawing-hoard ABCD, on which 
the original curve is placed. Fixed to each long side of this 
board is a metal rail, one, CE, having a plain surface, and the 
other, DF, a longitudinal groove of Y-shaped section. The second 
part is a smaller hoard, CHI, having three spherical feet, two 
of which run in the groove and the third on the plane rail. 
This arrangement permits free motion of the smaller hoard in the 
direction of the length of the larger one, i.e. parallel to the Y 
coordinate. The small hoard carries the paper on which the 
derived curve is traced by the machine. Attached to its edge 
are guides, J K L M, which hold the principal part of the 
mechanism, allowing it free motion in a right and left line. 
This part, shown in fig. 2, consists of a frame ABCD, at 
one corner of which is a pin, A, which serves as the vertical axis 
about which the rod P Q revolves in a horizontal plane. P Q has 
a slot in it, through which passes the pin R fixed to the rod S T. 
S T is controlled by guides E and F, so that it can only move in a 
direction parallel to 0 Y. 
Below the arm P Q, and fixed rigidly to it below A, is a small 
