Vibrations of Solids. 17 



there is an odd number of segments, there is a node in the 

 middle of the lath, and that in all cases the equal segments, 

 which we may call normal segments, are grouped symmetri- 

 cally in regard to the ends and middle of the lath. 



§ 3. The conditions of a lath free at both ends were studied 

 by the men above mentioned, and by Strehlke, and are strictly 

 comparable with those of the half-free lath. A quite free 

 lath always vibrates symmetrically about its geometric centre, 

 whether the material centre be at rest or in motion. When 

 the centre is at rest, and therefore forms a node, the free lath 

 might be supposed to have motions similar to those of two 

 half- free laths joined end to end. And so we might expect 

 that the same lengths of segments would be observed in the 

 quite free lath as in the half-free one. 



According to Seebeck, the distance from the end of a quite 

 free lath having natural nodes, when n + 1 is the number of 

 nodes, is got from the expressions : — 



1*322 



the distance of first node from end = -. — — ^ I, 



4^ + 2 ' 



, 4-9820 7 



t, „ second „ „ = ^j^h 



,,. , 9-0007 , 



» " " = 4^+2 ' 

 , , _ 4m — 3 j 



•» mth » » ~4^+2 



The last of these, giving the distance from the free end of the 

 quite free lath of the mth node, of course expresses the equality 

 of the distances from one another of all nodes after the third, 

 and so embraces the generalization i. in § 2 ; but it has no 

 direct bearing upon the lengthening of the last segment in 

 the half-free lath. 



From Seebeck's constants and general expression, adopting 

 for the sake of direct comparison my length of 280 millims. 

 for the quite free lath, we should have the numbers of fig. 2. 



These numbers agree remarkably well with" mine. But 

 what kind of agreement is it ? In the quite free lath having 

 two natural nodes, the end segment is equal to the end seg- 

 ment of the half-free lath of the same length having one arti- 

 ficial and one natural node, and so on for more numerous 

 nodes. If, therefore, we have a quite free lath of length I 

 vibrating with a node in the middle — that is, having an even 

 number of segments (say 2n) — and then clamp it at the central 

 node, we alter its condition of vibration in so far as we lengthen 

 the end segment, which was formerly one of the two middle 



Phil. Mag. S. 5. Vol. 9. No. 53. Jan. 1880. C 



