202.] PLANE MOTION. IO ; 



first try to make n i points of the given curve taken between these 

 limits coincide with as many points of the curve 



f(x) = Z?! sin x + BI sin 2 ^ H +B n ! sin ( i)x. (24) 



Then passing to the limit for = oc, the problem will be solved. 



201. Dividing the interval from x = o to x = ir into equal parts 

 and taking the i points on the given curve so that their abscissae 

 are 



TT 2ir 37T (rt I)T 



n n n n 



he curve (24) will contain these points if the following n i equations 

 are fulfilled : 



fl J " \ n . " . r . _ 2 7T 



_. . ^ 



= ^ lS in(-i)^+^sin(-i)^+...4-^ n -isin(-i)^-^. 



. . . (25) 

 These equations are sufficient to determine the i unknown con- 



stants 3 lt B* ., ^ n _i. 



202. To solve the equations we multiply them by indeterminate co- 

 efficients and add. The coefficients can be so selected as to make all 

 the unknown quantities disappear except one which is thus determined. 

 Thus, to find B m multiply each equation by twice the coefficient of 

 B m in this equation, viz. the first equation by 2 sin (mir/ri), the second 

 >y 2 sin (2 mir/n), etc., the last by 2 sin[( i)mir/ri]. 

 | After adding, the factor of B k will be 



, 7T . 7T . 2 7T . 2 7T , 



B k = 2\ sm/fc-smw + sm smm h 



n n n n 



+ sin k (n i) -sin m(n i) - 



