of varying Elasticity. 265 



ki each of the i surfaces of separation there are two equa- 

 tions of the types (21) and (22). At the outmost surface 

 r=a, neglecting atmospheric pressure as before, we have 



K--tt i )B + 2m i A;-^'A / J - = 0; . . (23) 



and at the inmost surface r=b a similar equation, writing b 

 for a and dropping the suffix. If the cylinder be not hollow 

 this last equation does not exist, but in its place we obviously 

 have A! = 0. Thus there are 2i + 2 or 2i + 1 equations accord- 

 ing as the cylinder is hollow or not, to determine the constants 

 of types A and A' in terms of B. Of these constants each 

 medium possesses two, except when the cylinder is solid, 

 when the central medium has only one constant, viz. A. 

 Thus, whether the cylinder be hollow or not the equations are 

 the same in number as the constants, which thus may all be 

 directly expressed in terms of B. The values of the constants 

 can be at once written down under the form of determinants. 

 To determine B, we have 



F = 7r[(a 1 2 -6 2 )R + (a 2 2 -« 1 2 )R 1 + ...(a 2 -a, 2 )R i -]- 



For the special case — = constant for all values of s, of 

 n s 



which we have previously seen the importance, a very simple 



solution holds. In fact it is obvious from (21), (22), and 



(23) that all our conditions are then satisfied by 



A s — A s _i — . . . — 0, 



-aB ) 



n^ ' - C 24 ) 



in our previous notation, a being, in accordance with this 

 hypothesis, the same for all the media. In this case the equa- 

 tion for B is simply 



F = 7rB[(a 1 2 -& 2 )M + (a 2 2 -a 1 2 )M l + ... + (a 2 -a i 2 )M i ],...(25) 



where M, M 1? &c. denote Young's moduli for the several media. 

 Also the solution (19) applies to all the media. 



The equations (21) and (22) have been established inde- 

 pendently of the absolute thickness of the layers : thus, under 

 certain limitations to be presently considered, they may be 

 supposed to hold when the thickness is indefinitely reduced, and 

 thus in the limit to apply to a continuously varying medium. 

 Thus, dropping the suffixes and writing r for a s+] , we get in 

 place of the constants of types A and A' certain functions of 



Phil Mag. S. 5. Vol. 22. No. 136. Sept 1886. T 



