OF ARTS AND SCIENCES. 53 



It is well known that for any continuous curve, however irregular, 

 extending from x = — n to x = n, the relation of any y to the corre- 

 sponding x may he written as follows : — 



y = (f> (x) = % b + b x cos x + b 2 cos 2 x + . . . -f b m cos m x -f . . . 

 -f «! sin x + «2 sin 2 a; + . . . + a m sin m a: -f . . . 



The length of this series of terms may be, and usually is, infinite, 

 but frequently a comparatively small number of terms will express 

 the required relation with sufficient accuracy, when the values of 

 the coefficients b , b u a u etc. are known. I used eleven terms, 

 six containing b coefficients and five containing a coefficients, and 

 the labor of determining the coefficients was considerable. The 

 general expression for any b coefficient is 



b m = I I <K«) cos mad a, 



and for any a coefficient 



a m = - I 4>(a) sin m a d a, 



-It 



in which expressions a is any variable beginning with the value 

 — 7r, and increasing regularly to +7r, and <£(a) bears to a the same 

 relation that <p (x) bears to x. The process of obtaining the value 

 of any coefficient is mainly graphical. To illustrate, let us take the 

 case of bo. The value of cos 2 a was found for fifteen values of a, 

 beginning with a = —tt, and ending with a — +ir. The corre- 

 sponding values of <£(a) were obtained from the heavy curve already 

 described in Figure 2 by measurement of the ordinates correspond- 

 ing to the chosen values of a. Then the product <£(«) cos 2 a was 

 taken for each of the chosen values of a, and, anew base line extend- 

 ing from —tt to +7T having been laid off, these products gave the 

 values of the ordinates at the chosen fifteen points along this base 

 line. Through the tops of the ordinates a curve was drawn, and 

 the area between this curve and the base line was measured by 

 means of a planimeter. This area gave the value of the definite 



integral 



J—TT 



<f)(a) COS 2 a d a. 



