270 Mr. E. Howard Smart on the Third-Order 



ZnS above, as well as in NaCl in the previous paper, are due 

 to this cause. The diamond appears to be unique in that 

 there is only one kind of atom and that the inner ring does 

 not contain one or two electrons. I am not aware that any 

 other crystal has been studied experimentally where none of 

 the atoms have, according to the theory, one or two electrons 

 at the centre. 



Fig. 5 is a photograph of a model of the diamond, and inav 

 be found of assistance in supplementing figs. 3 and 4. 



Note. 



Since this paper was communicated a considerable number 

 of crystals on the isometric system has been worked out, and 

 experiment found to agree with theory. These include zinc 

 and copper. 



XXIV. The Third-Order Aberrations of a Symmetrical 

 Optical Instrument. By E. Howard Smart, M.A., 

 Head of the Mathematical Department, Birkbeck College, 

 London *. 



THE five third-order aberrations of a symmetrical optical 

 instrument, commonly associated with the name of 

 Von Seidel, have been frequently discussed. But the mode 

 of presentation often leaves something to be desired from 

 the practical optician's point of view, direction cosines of 

 rays and the like being to him of inferior importance com- 

 pared with angular aperture and field of view. Sometimes 

 even the whole subject is treated in general terms, the 

 constants of the instrument not being considered ; and 

 occasionally there is some obscurity regarding the relations 

 between the several errors. 



In this paper an attempt has been made to effect some 

 improvement in these respects, and at the same time to 

 indicate a method by which the investigations could be 

 extended so as to deal with the fifth-order corrections. 



Let Ci (fig. 1) be the centre and 0; the vertex of the ith of a 

 system of coaxial spherical surfaces, and let this surface 

 separate media of refractive indices fi i _ 1 and fi v Let r { be 

 the radius of curvature of the surface considered positive 

 when the surface is convex to the incident light. Take 0; as 

 origin and a pair of rectangular tangents at 0;, and the axis 

 of symmetry as coordinate axes. 



* Communicated bv the Author. 



