334 Dr. C. Chree on Longitudinal 



u obtained as a second approximation by Lord Eayleigh." I 

 further said, " We do not think, however, that his proof affords 

 any means of judging of the degree of accuracy of the result, 

 as it is founded on a more or less probable hypothesis and 

 does not profess to be rigid/' I subsequently learned that 

 Lord Eayleigh did not admit any want of rigidity in his 

 proof, and it appears without modification in the second 

 edition of his Treatise on ' Sound/ I much regret having to 

 differ from so eminent an authority, but I have not altered 

 my original opinion. 



In (B) I reached the more general result 



k=p(V/p)i(l-i P YK 2 ), (3) 



where k is the radius of gyration of the cross section of the 

 rod about its axis. This was established by a strict elastic 

 solid method for an elliptic section, and in a somewhat less 

 rigid way for a rectangular section. (A) and (B) were con- 

 fined, like the investigations of Lord Eayleigh and Professor 

 Pochhammer, to isotropic materials. 



In (C) I considered the more general case of an seolotropic 

 bar whose long axis was an axis of material symmetry, and 

 found by strict elastic solid methods that (2) still held for 

 a circular section, if E denoted Young's modulus for stress 

 along the length of the bar, and rj Poissou's ratio for the 

 consequent perpendicular contractions. Further, applying 

 Lord Eayleigh/ s method, modified in a way I deem necessary, 

 I obtained (3) for any form of cross section. 



§ 3. Since the publication of (( J) Mr. Love has discussed the 

 subject in vol. ii. of his l Treatise on Elasticity/ On his p. 1 19 * 

 he refers to (2) as " first given by Prof. Pochhammer . . . 

 and afterwards apparently independently by Mr. Chree/'' 

 Again, in the new edition of his ' Sound ' Lord Eayleigh, 

 after deducing (2), says a A more complete solution. . . has 

 been given by Pochhammer ... A similar investigation has 

 also been published by Chree.'"' 



In view of these remarks, I take this opportunity of stating 

 explicitly : — 



1. That Pochhammer's work was wholly unknown to me 

 until the appearance of Love's l Elasticity/ 



2. That my method of solution in (A) is essentially different 

 from Pochhainmer's, while the methods in (B) and (C) are 

 absolutely different from his. The method of (A) agrees 

 with Pochhammer's in employing the equations of elasticity 

 in cylindrical coordinates. After obtaining, however, — as is 



* The preface, p. 13 ; describes the result as " obtained independently " 

 by me._ 



