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



Having obtained the expressions for the normal co- 

 ordinates, we may write out the complete values of the 

 deflexions, ?/, and the velocities, y, at any point x and time t 

 by the relations in (14) and (15) between the normal co- 

 ordinates and these quantities. As a preliminary step we 

 will write out in full the values of he, ks, n S) and e s as s takes 

 in succession the values 1, 2, 3 &c. 



If 5 is odd, by (18) 



(39) 



7 4H • 1 - hl • h - lh • 



tip 3 5 



h - - 1 



. . fls — 



S 



If even 





/, 2 =A 4 = /, 6 = &c. = 0. 





If 5 is odd 





h 2 h 2 



■■ iks - k+ s m P 



(40) 

 If even 



^2 = & 4 == & 9 = &C = k. 



By (13), (SG^and (27) 



7T /l\ n a n x //M . 



= TV7 ;B! = i ; •••^=7 • • • • (41) 



p 



k s (o s 2 % 



(42) 



Slne «~ [( 5 v-« 2 )+^> 2 ] §; ° oses ~ [(sV-» 2 )+y© 2 ] L 



The permanent periodic portion of the motion of any 

 point of the string in a uniform magnetic field is therefore 

 (s being odd), by (14), (25), and (42), 



4HI f cos («£ — 6i) . 7T^ cos (art — e 3 ) . 3ttx 



j : ii sin- — -4- — sin ■ 



irp Ll[(V-« 2 ) 2 + ^iV]" l 3[(3V-^ 2 ) 2 + W]* l 



. S7TX^\ 



COS ((Ot — €) 



It is to be remarked that the coefficients of the terms in 

 this equation have particular numerical values because of the 

 original assumption that the string is immersed in a uniform 

 magnetic field. This causes the curve for constant current 

 to be the arc of a circle. At each and every point of the 

 string, corresponding to a single value of x, the motion of 

 that point is evidently a simple harmonic function of the time, 

 having a period agreeing with that of the impressed force, 

 but differing from it in phase, since all the simple harmonic 



