Vibration of the Gravest Mode of a Thin Rod. 263 

 This gives for the uniform rod, ^ 4 = 12*46 + 2*35A, . . . (i a) 



,, rod, tapering 

 to a knife-edge, /* 4 = 39-37 + 2-39\ . . . (i&) 



Secondly, let us assume y = 17 ?_, where 77 is the displace- 

 ment of" the free end. I 



This is likely to give better results for the rod tapering to 

 a knife-edge than the first assumption, as only one of the 

 end conditions now becomes indeterminate. 



Proceeding in the same way as before, we find 



P.E. due to bending 



-^•^[i-GfJ], 



P.E. due to the tension 



= P a3Wr}2 [l5^TTc]' 



Hence J + eV* _i T, \ f 4 2 //c ' 1 



[/"(l + Z/c) 4 ] +A Ll5 SU + Z/cJ 



^M (i + Z/c) 



1 1 Z/< 



10 121 + Z/c 



+ 3 



#\. 



6 1 + c/Z 



This gives for the uniform rod, ^ 4 =20 + 2\66\ . . . (iia) 

 „ „ „ rod, tapering 



to a kuife-edge, //, 4 = 30 + 2'66\ . . . (ii b) 



To obtain some idea of the closeness of the approximation, 

 (i a), (i b), (ii a), (ii b) have been graphed in figures 2 a, 2 6, 

 and compared with the more accurate results obtained in the 

 two papers already referred to. It should perhaps be pointed 

 out in passing, that for the uniform rod when \ = 7. fi l should 

 be 29*2 and not 32*1 as given in Mr. Berry's paper. With 

 this correction, wc see that the lower Rayleigh curve for the 



