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XXI. Mechanical Integration of the Product of two Functions. 

 By William Sutherland, M.A., B.Sc* 



IN a communication to the Royal Society (February 3, 

 1876), "On an Instrument for Calculating f <£(#)^r(#)tf#, 

 the integral of the product of two Functions/' Prof. Sir 

 William Thomson has shown how, by the use of Prof. James 

 Thomson's disk, globe, and cylinder integrating machine, the 

 integral of the product of two functions can be found. The 

 operations involved consist, first, in plotting the curve y = ^r(#), 

 then in making the instrument, by a rather difficult attach- 



ment, yield a trace of the curve y = I y^(d)dx, then in plot- 

 ting the curve y=<j>( w ) : these two latter curves have to be 

 wrapped round one or two cylinders. When the cylinders are 

 caused to revolve, a pointer capable of moving parallel to the 

 axis of y has to be kept on each curve, the motions of the two 

 pointers being communicated to the disk and globe respectively 

 of the integrator : the amount of the angular movement of 

 the cylinder gives the integral. 



In view of the very important part that the analysis of an 

 arbitrary function into its harmonic constituents is destined 

 to play in extracting law out of the immense mass of physical 

 and chemical measurements that are being accumulated, it 

 seemed to be worth while to look for a simpler method of 

 obtaining J <j>(x)yfr(x)dx than the above. The following may 

 be found to be such, and seems likely to be capable of more 

 immediate application ; for the disk, globe, and cylinder 

 integrator, despite its kinematical elegance, has not yet come 

 into general use. 



The following method is merely a mechanical realization of 

 the operations which Fourier so carefully describes in his 

 * Analytical Theory of Heat,' art. 220, chap. iii. sect, vi., in 

 order to give as concrete an idea as possible of the meaning of 

 the coefficients in his expansion. u We see by this that the 

 coefficients a, b, c, d, e,f which enter into the equation 



■^7r<£(#) = a sin m + b sin 2# + c sin 3# + d sin 4# + &c, 

 and which we found formerly by successive eliminations are 

 the values of the definite integrals expressed by the general 

 term j sin ix$>(x)dx, i being the number of the term whose 

 coefficient is required. This remark is important, because it 

 shows how even entirely arbitrary functions may be developed 

 in series of sines of multiple arcs. In fact, if the function 

 4>(a) be represented by the variable ordinate of any curve 

 * Communicated by the Author. 



