1897.] Mr Newall, On the marks made by stars etc. 271 



than the average intensity within the boundary. Note added 

 1897, Mar. 15.] 



The effects of spherical aberration would be distinguishable 

 from the diffractional intensification near the boundary by the 

 fact that the aberration effects would be different inside the 

 focus from those outside the focus, but the diffractional effects 

 would be the same on opposite sides of the focus. 



(3) Theorems on the contacts of spheres. By Mr W. M C F. Orr, 

 M.A. 



Few, if any, of the following theorems are new. Theorem (1) 

 is equivalent to the well-known theorem that all the generating 

 spheres of one system of a cyclide touch all the generating spheres 

 of the other system. Theorem (5) is proved analytically by Casey 

 (Proceedings Royal Irish Academy, Vol. ix.). I believe they are 

 not generally known, however, and it may be worth while to bring 

 them together. 



1. The problem to describe a sphere touching four spheres 

 may admit of an infinite number of solutions. For any four 

 tangent planes from a point to a sphere are touched by an infinite 

 number of spheres ; by inversion from any point the four planes 

 become four spheres touched by an infinite number of spheres. 



2. Any two spheres and their inverses with respect to a third 

 are touched by two infinite families of spheres. For if A, B be 

 two spheres and A', B' their inverses with respect to another 

 sphere 0, the spheres which touch A and A' are divided into two 

 families each doubly infinite, all the members of one of which 

 cut orthogonally. Any sphere of this family which touches B 

 must therefore touch B' and belongs to one or other of two singly 

 infinite families according as the natures of its contacts with A 

 and B are the same or different. 



3. Any three spheres and their inverses with respect to a 

 fourth are touched in common by eight spheres and in general 

 by no more. For HA, B, G be three spheres and A', B ', G' their 

 inverses with respect to another sphere 0, of the sixteen spheres 

 which can be described touching A, B, G, A' (in case the solu- 

 tions of that problem are determinate), eight cut orthogonally 

 and therefore touch B' and C 



4. The sixteen spheres which can be described to touch four 

 given spheres (in case the solutions of that problem be determinate), 

 consisting of eight pairs of conjugates, the members of each pair 

 being inverse to each other with respect to the sphere orthogonal 

 to the original four, any two pairs of conjugates are touched by 

 two infinite families of spheres. This follows from theorem (2). 



