176 Mr. W. Sutherland on the Mechanical Integration 



whatever, whose abscissa extends from #=0 to #=7r, and if 

 in the same part of the axis the known trigonometrical curve 

 whose ordinate is y = sin x be constructed, it is easy to repre- 

 sent the value of any coefficient. We must suppose that for 

 each abscissa x to which corresponds one value of (f>(x) and 

 one value of sin x, we multiply the latter value by the first, 

 and at the same point of the axis raise an ordinate equal to the 

 product (})(x) sin x. By this continuous operation a third 

 curve is formed, whose ordinates are those of the trigono- 

 metrical curve reduced in proportion to the ordinates of the 

 arbitrary curve which represents <\>(x). This done, the area 

 of the reduced curve taken from x = to x=7r gives the exact 

 value of the coefficient of sin #." 



To take the more general case, let it be required to find 



1 cj)(0)yjr(0)d0. Plot the two curves whose polar equations 



are r = (f>(0), r = i|r(^), using the same pole and the same initia 

 line for angular measurement in both cases. Let Si and S 2 h 

 the points on the two curves corresponding to any value of t 

 (see figure) ; then 



^(0)^(0) = OSi . OS 2 . 



Describe a circle of radius R 

 passing through S x and S 2 ; 

 through O draw the tangent T 

 to this circle, then 



OT 2 =OS!.OS 2 . 



Now if the arm OSiS 2 is turned 

 into any other position, cutting 

 the curves in two fresh points S'i 

 and S' 2 , and if the circle of con- 

 stant radius R is again brought 

 into position so as to pass through 

 these points, the new length of 

 the tangent 01 v is always such 

 that OT'^OS'i . OS^: therefore 

 generally OT 2 = yfr(0)(f>(0). 



In OSiS, mark off OS 3 = OT ; 

 then if, as OSiS 2 is revolved 

 from the initial to the final position, or from that given by 

 — ol x to that by = oc 2 , the motion of T is by any mechanical 

 device communicated to S 3 unaltered, the locus of S 3 is a curve 

 such that always O8l = cj>(0)f(0) ; 



.-. ^<f>(0m0)cW=^ 2 O8lcW 



