xii Proceedings. [December i6th, rgo2. 



cross the optic axis of a lens system " would frighten off even a 

 practised computer," he pointed out that this difficulty is largely 

 due to the method adopted, according to which one has to find 

 the intersection points of rays, which method is generally tedious 

 and breaks down for lateral rays passing through imperfectly 

 designed lenses. A very much simpler method would be to 

 find the paths in a system of lenses of various rays of a beam of 

 light, to compute their focal lengths accurately, and if these foci 

 do not fall at the same distance together, — in which case the 

 focal region is reduced, as it should be, to a point (the apex of a 

 cone), — to treat them as if they were tangents of a caustic. 

 This process is only correct for those rays which may be 

 supposed to pass through elementary radial apertures. The 

 paths of those rays which may be supposed to pass through 

 elementary tangential apertures coincide with the paths of those 

 just mentioned, but the focal lengths of the two systems of 

 rays are not necessarily the same. The tangential rays must, 

 however, be looked upon as tangents to the caustics of the 

 lateral rays. With the help of this explanation and a few 

 simple formulae given in the paper, it is possible to draw the 

 radial and lateral caustics for two sheets of light placed normally 

 to each other, and, with the help of the figures thus obtained, it 

 is then a simple matter to obtain an enlarged representation of 

 the focal region of a beam of light. This focal region, as already 

 mentioned, should be the point of a cone. 



The following paper was communicated by Mr. R. L. 

 Taylor, F.C.S.:— 



A Simple Form of Vernier Microscope. 



By A. A DAMSON, A.R.C.S. 

 This apparatus is specially devised to suit the elementary 

 student in a physical laboratory who is familiar with the use of 

 the vernier, and who wishes to calibrate or determine the bore of 

 a glass tube by measuring the length of a mercury thread 

 within it. The ordinary form of vernier, or micrometer, 

 microscope used for this purpose is expensive, and usually too 



