OF ARTS AND SCIENCES. 53 



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

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

 sponding X may be written as follows : — 



y = (f) (x) = ^ b^ + bi cos X -\- b^ cos 2 x + • • • + ^m cos mx -{- . .'. 

 + Ui siu X + aj sin 2x + . . . -\- a„ sin m x -}- . . . 



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 bo, bi, ai, 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 



^m = I / ^(a) cos mad a, 

 and for any a coefficient 



sin m a d a, 





in which expressions a is any variable beginning with the value 

 — TT, and increasing regularly to +7r, and ^(a) bears to a the same 

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

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

 case of Sj- The value of cos 2 a was found for fifteen values of a, 

 beginning with a = — tt, and ending with a = +7r. 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 cf) (a) cos 2 a was 

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

 ing from — TT to -{-TT 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 



Hi 

 J — jj- 



■tr 



(P{a) COS 2 a d a. 



