ANALYSIS INTO FRICTIONAL SINUSOIDS. 105 



The equation can be written 



e".y = (h.cos-^,t + a2.eos—-t+ ... (5) 



+ b,Mnp + b,sm^t+ . . . 



From this the coefficients Oi, a.2, . . . , hi, hi, . . . can be obtained by 

 considering e".y as a curve to be subjected to the harmonic analysis. The 

 coefficients are therefore 



2 T' /.2;rx 



^i= Y>fe".ij. COS {ijr,t).dt 



(6) 



h,= ^,f".yMn(i^p}dt (7) 







(t=l, 2, 3, . . .) 

 A convenient form of (5) is 



c".t/ = Ci.sin(jJ^-gi) + C2.sin(^^,— ?.,)+ ... (8) 



where 



c = l/^M-F and tan q = — ^. (9) 



When the wave length is divided into m equal parts h (that is, 

 T'='mh), the time axis has the equidistant points 



U,U,U, . . . t„ . . . L.,^0,h,2h, . . . jh, . . . {m-l)h, 



for which the ordinates multiplied by the respective elements of friction 



become 



e-'^.yo, e'\y„ C'-.y,, . . . e"'.?/„. . • e"-\y„,-,. 



With these finite values the coefficients become 



j — m—l 



tti = — "^ e''''.y,.cos ijh, 

 m ^_ 



(10) 



y = m — 1 



2 »^ 

 6. = — >. e''*. t/,.cos ijh. 

 m ^_- 



,=0 



