54 Prof. Gayley, On a system of two tetrads [Feb. 13, 



(2) On the isotropic elastic sphere and spherical shell. By 

 C. Chree, M.A., King's College. 



In a paper communicated to the Society in 1887 the author 

 gave a mathematically complete solution of the equations of equi- 

 librium for an elastic solid sphere in polar coordinates. The 

 expressions for the elastic displacements contained arbitrary con- 

 stants to be determined by the surface conditions. These constants 

 were, however, found explicitly only in the case of normal forces 

 over a solid sphere. The first object of the present paper is to 

 determine the arbitrary constants for all forms of surface forces 

 over both surfaces of a shell. The expressions for the typical 

 displacements are given explicitly. The forms taken by the dis- 

 placements, strains and stresses when the shell becomes very thin 

 are deduced. The order of magnitude of the stress usually assumed 

 to vanish in theories of thin shells relative to the other stresses 

 is determined, and their mode of variation along the thickness of 

 the shell is illustrated by means of curves. The problem when the 

 two surfaces of a shell suffer any given arbitrary displacements is 

 also solved explicitly, the form taken by the results when the shell 

 is very thin being more particularly considered. Several other 

 applications are made of the solution to cases of physical interest. 



(3) On a system of two tetrads of ciixles; and other systems 

 of two tetrads. By Prof Cayley. 



The investigations of the present paper were suggested to me 

 by Mr Orr's paper, " The Contacts of certain Systems of Circles.*" 



1. It is possible to find in piano two tetrads of circles, or say 

 four red circles and four blue circles, such that each red circle 

 touches each blue circle : in fact counting the constants, a circle 

 depends upon 3 constants, or say it has a capacity = 3 ; the 

 capacity of the eight circles is thus = 24 ; and the postulation or 

 number of conditions to be satisfied is = 16 : the resulting capacity 

 of the system is thus prima facie, 16 — 24 = 8. It will, however, 

 appear that in the system considered the true value is = 9. 



2. The prima facie value of the capacity being = 8, we are 

 not at liberty to assume at pleasure three circles of the system. 

 And in fact assuming at pleasure say 3 red circles, then touching 

 each of these we have 8 circles, forming g^ 8 . 7 . 6 . 5, = 70, tetrads 

 of circles : taking at random any one of these tetrads for the blue 

 circles, the remaining red circle has to be determined so as to 

 touch each of the four blue circles, that is by four instead of three 

 conditions ; and there is not in general any red circle satisfying 

 these four conditions. But the 8 tangent circles do not stand to 

 each other in a relation of symmetry, but form in fact four pairs 



* Proc. Camb. Phil. Soc, Vol. vii. 



