266 Prof. Sir W. Thomson on Calculating [Feb. 3, 



IV. c< On an Instrument for calculating C\ <p (w) \|/ (#) efo? Y the Inte- 

 gral of the Product of two given Functions." By Prof. Sir 

 W. Thomson, LL.D., F.R.S. Received January 28, 1876. 



In consequence of the recent meeting of the British Association at 

 Bristol, I resumed an attempt to find an instrument which should super- 

 sede the heavy arithmetical labour of calculating the integrals required 

 to analyze a function into its simple harmonic constituents according to 

 the method of Fourier. During many years previously it had appeared 

 to me that the object ought to be accomplished by some simple mechanical 

 means ; but it was not until recently that I succeeded in devising an 

 instrument approaching sufficiently to simplicity to promise practically 

 useful results. Having arrived at this stage, I described my proposed 

 machine a few days ago to my brother Professor James Thomson, and he 

 described to me in return a kind of mechanical integrator which had 

 occurred to him many years ago, but of which he had never published any 

 description. I instantly saw that it gave me a much simpler means of 

 attaining my special object than any thing I had been able to think of 

 previously. An account of his integrator is communicated to the Boyal 

 Society along with the present paper. 



To calculate f f (at) ^(ayix, the rotating disk is to be displaced from a 

 zero or initial position through an angle equal to f*<j5(V)c^, while the 

 rolling globe is moved so as always to be at a distance from its zero posi- 

 tion equal to i/>(V). This being done, the cylinder obviously turns through 

 an angle equal to y ' <p(oa) ^(x)doc, and thus solves the problem. 



One way of giving the required motions to the rotating disk and rolling 

 globe is as follows : — 



On two pieces of paper draw the curves 



y=\ *<p(jv)dx 9 and y—i>{x). 



Attach these pieces of paper to the circumference of two circular cylin- 

 ders, or to different parts of the circumference of one cylinder, with the 

 axis of oo in each in the direction perpendicular to the axis of the cylinder. 

 Let the two cylinders (if there are two) be geared together so as that their 

 circumferences shall move with equal velocities. Attached to the frame- 

 work let there be, close to the circumference of each cylinder, a slide or 

 guide-rod to guide a movable point, moved by the hand of an operator, 

 so as always to touch the curve on the surface of the cylinder, while the 

 two cylinders are moved round. 



Two operators will be required, as one operator could not move the 

 two points so as to fulfil this condition — at all events unless the motion 

 were very slow. One of these points, by proper mechanism, gives an 



