262 Mr Orr, On the Contact Relations of certain [Nov. 23, 



(2) The Effect of Flaws on the Strength of Materials. By J. 

 Larmor, M.A., St John's College. 



The effect of an air-bubble of spherical or cylindrical form in 

 increasing the strains in its neighbourhood was examined ; and it 

 was suggested that the results might be of practical service in 

 drawing general conclusions as to the influence of local relaxations 

 of stiffness of other kinds. In particular, it is shewn by the aid 

 of the hydrodynamical form of St Venant's analysis, that a cavity 

 of the form of a narrow circular cylinder, lying parallel to the 

 axis of a shaft under torsion, will double the shear at a certain 

 point of its circumference; and the effect of a spherical cavity 

 will not usually be very different. For a cylindrical cavity of 

 elliptic section, the shear may be increased in the ratio of the 

 sum of its two axes to the smaller of them, this ratio becoming 

 infinite in accordance with known theoretical principles for the 

 case of a narrow slit. It is assumed in the analysis that the 

 distance of the cavity from the surface of the shaft is considerable 

 compared with its diameter, so that the influence of that boundary 

 may be left out of account in an approximate solution 1 . 



The results will however also give the effect of a groove of 

 semicircular or semi-elliptic section, running down the surface of 

 the shaft, provided the curvature of the surface is small compared 

 with the curvature of the groove. 



(3) The Contact Relations of certain Systems of Circles and 

 Conies. By W. M C F. Orr, B.A., St John's College. 



(Abstract.) 



The following theorem is first established: — If four circles 

 X, Y, P, Q, in a plane or on a sphere, are such that a circle can 

 be drawn through one of each pair of intersections of X with P, 

 X with Q, Y with P, and Y with Q respectively, (and therefore 

 another circle through the remaining four intersections of the 

 same circles,) the eight circles which touch X, Y and P, and the 

 eight which touch X, Y and Q, can be arranged in sixteen groups 

 of four circles, each group consisting of two touching X, Y and P, 

 and two touching X, Y and Q, such that each group has two 

 common tangential circles besides X and Y. 



A similar result is of course true for groups of circles touching 

 P, Q and X, and P, Q and Y respectively. 



The above relation of condition is triply satisfied by the four 

 circles that form any Hart-group of circles touching three others 

 (restricting the title Hart-group to the eight groups that are 

 analogous to the inscribed and escribed circles of a triangle). 



1 See Phil. Mag., Jan. 1892. 



