218 Dr. A. C. Crehore on the Theory of the 



when the frequency of the forced motion is very small com- 

 pared with the natural fundamental period, and the damping 

 is not excessive, the curve of the string, at the instant of 

 maximum electromotive force, should be the same as if a 

 direct current flowed in the string. The equation shows 

 this to be the case. Putting £ = 0, when the e.m.f. (24) is 

 a maximum, we have 



_ 4HI r n^-co 2 . ttx 3V-0) 2 . Zttx 



y - ^ / rL(ni 2 -« 2 ) 2 +^w sm i + 3[(3w-® 2 ) 2 +^y] sm i 



whence, making o> small compared with n x and neglecting 

 h s (o as compared with 7ij 2 , we have 



4HI f . irx 1 . ?>irx 1 . 07TX \ ,._. 



^ = 7r^?( sm T + y sin -r + p sm -r •"•••) • ( 48) 



7rpiii 

 and by (7) and (13) 



4HI 32 



IT' 



Hence the form of the string agrees with (9), which is the 

 approximate equation of the circular arc. 



As the frequency of the impressed force increases, other 

 things remaining the same, the form of the string at its 

 maximum deflexion changes. For example, let us increase the 

 impressed frequency in (43) so far as to reach the first critical 



IT 



frequency, so that a> = rc 1 . Then sine 1 = l and €j= ~-, and 



cos (cot — 6^ = sin cot, which becomes zero when the time is 

 zero and e.m.f. a maximum. The harmonics present are the 

 only elements which prevent the string from taking the form 

 of a straight line at this time. To find the curve at the 



77" 



maximum deflexion, put g>£= — . If the damping-factor is 



so small that k s co may be neglected in comparison with 

 Z 2 n^ — g> 2 , and h b <o with 5 2 n-f— co 2 , &c, all terms but the first 

 in (43) are negligible, and 



4HI . ttx /inx 



^=^ sm T W 



The curve at maximum deflexion has now changed to that of 

 a simple curve of sines from that of a circular arc. 



