52 
[719 
719. 
SUGGESTION OF A MECHANICAL INTEGRATOR FOR THE 
CALCULATION OF {(Xdx + Ydy) ALONG AN ARBITRARY 
PATH“. 
[From the Messenger of Mathematics, voi. vii. (1878), pp. 92—95; British Association 
Report, 1877, pp. 18—20.] 
I consider an integral J(.Xdx + Ydy), where X, Y are each of them a given 
function of the variables (so, y) ; Xdx + Ydy is thus not in general an exact differential ; 
but assuming a relation between (x, y), that is, a path of the integral, there is in 
effect one variable only, and the integral becomes calculable. I wish to show how 
for any given values of the functions X, Y, but for an arbitrary path, it is possible 
to construct a mechanism for the calculation of the integral : viz. a mechanism such 
that, a point D thereof being moved in a plane along a path chosen at pleasure, the 
corresponding value of the integral shall be exhibited on a dial. 
The mechanism (for convenience I speak of it as actually existing) consists of a 
square block or inverted box, the upper horizontal face whereof is taken as the plane 
of xy, the equations of its edges being y = 0, y= 1, x=0, x = l respectively. In the 
wall faces represented by these equations, we have the endless bands A, A', B, B' 
respectively; and in the plane of xy, a driving point D, the coordinates of which are 
(x, y), and a regulating point R, mechanically connected with D, in suchwise that 
the coordinates of R are always the given functions X, Y of the coordinates of Df ; 
the nature of the mechanical connexion will of course depend upon the particular 
functions X, Y. 
This being so, D drives the bands A and B in such manner that, to the given 
motions dx, dy of D, correspond a motion dx of the band A and a motion dy of 
* Read at the British Association Meeting at Plymouth, August 20, 1877. 
+ It might be convenient to have as the coordinates of R, not X, Y but f, -rj, determinate functions of 
X, Y respectively.