342 Dr. 0. Chree on Longitudinal 



From the nature of the elastic solid equations the expres- 

 sions for the displacements must be of the same general form 

 as for isotropy, so that we may still apply the formulae (6) 

 for a, ft, y. Doing so, and following exactly the same pro- 

 cedure as in the case of isotropy, we obtain from (16), for 

 any shape of section, 



(E 3 p - k*plp) (C + ...) = - *V(%i A^ 2 + *AV) + • . -, (19) 



and from (17) and (18) as first approximations 



A 1 /(%3E 3 /E 1 )=B 1 '/( %3 E 3 /E 2 ) = -p(C +...). . (20) 



Thence we obtain at once 



k*p=r>E 3 {l-p*JZ 3 (^-W+ ?gV)} • (21) 

 Employing (15) we give this the more elegant form 



*V -j*E»{ l -f ( Vsi W + % 2 V) \ , 



whence 



k=p(K 3 / P )Hl-i P *(vnW + V S 2W)\. ■ (22) 

 For a circular section of radius a 



k=p(E i /p)i{l-±p*a*( V51 * + V32 *)}. . . (23) 



For a rectangular section 2a x 2b ; the side 2a being parallel 

 to the #-axis, 



k=p(E 3 /p)h{l-ip*( a * V3] ? + b\ 3 /)\. . . (24) 



For a given size and shape of rectangle, the correction to 

 the first approximation is largest when the longer side is that 

 answering to the larger Poisson's ratio (for traction along 

 the rod). Possibly experimental use might be made of this 

 result in examining materials for seolotropy. 



If the material, though not isotropic, be symmetrical in 

 structure round lines parallel to the length of the rod, 



V3 i =Vs2 = V, say, 

 and writing E for E 3 in (22) we reproduce the result (60) of 

 paper (C). 



The results (22), (23), and (24), so far as my knowledge 

 goes, are absolutely new. 



Extension of Earlier Results. 



§ 9. My paper (A), like the corresponding investigation of 

 Pochhammer, dealt only with a solid circular cylinder ; but 

 the same method is applicable to a hollow circular cylinder. 

 For greater continuity I shall employ in the remainder of 

 tbis paper the notation of paper (A) . 



