90 Mr. E. R. Darnley on the Transverse Vibrations 

 property of reciprocal continuants, 



A*=2A«(*»A»-;+ViA.-i) 



and <f>nAn— f 2 «-iA }l -i = 



is the period eqnation for n bays fixed in direction at one 

 extremity. Thus the periods for the 2n bays are the smne 

 as those for the system divided at its middle point when 

 (1) the middle point is simply supported and (2) the beam 

 is fixed in direction at its middle point. 



In the first case the modes of vibration are skew sym- 

 metrical about the middle point, and in the second case they 

 iir^ symmetrical. 



When the number of bays is odd and equal to 2n — 1, by a 

 theorem in continuants 



A27i-1= An 2- ^7i 2 A 2 n-l, 



and the period equations are 



A»±.^»An-l = 0. 



Taking the upper sign, a single determinant is obtained 

 which differs from /\ n only in replacing the constituent 

 W>n-i + #») by (<£«-i -t-(/>,z + ^,0- Since <j> n + yjr n = (f)(lKl n ), 

 this equation reduces to 



A(/i, 4, ../«-i, IQ=0, 



and represents shew symmetrical vibrations, in which, of 

 course, the middle point is a node. 



Taking the lower sign, the equation gives the symmetrical 

 vibrations. It lias been shown in special cases that the 

 lowest period occurs among the symmetrical vibrations. 



§ 12. The case when all the bays are equal, the supports 

 remaining simple, requires to be considered separately. Let 

 there be n bays each of length I, so that nl is the length of 

 the whole beam. 



The results for one and two bays indicate that solutions 

 may be expected in the neighbourhood of the poles. Now 



the limit of y when KZ=S7r is ( — l) s , so that the period 

 equation can evidently be satisfied at all the poles, and the 

 least of these solutions is K= y • 



This is the well-known case in which the beam vibrates 



