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MR. T. Y. BAKER AND PROF. L. N. G. FIEON : LONGITUDINAL 
equivalent Gaussian system the ray and transverse magnifications are identical and 
agree with the transverse magnifications in the actual system. 
Finally the intercept of the incident ray on the leading principal plane of a system 
will usually be denoted by y. This is taken by some authors as the argument of the 
development in series, but differs only by a factor from tan a 0 or tan /3 2n . 
In many cases it will be convenient, in order to avoid unnecessarily large 
suffixes, to condense a system of surfaces or lenses, affecting quantities referring to 
the system itself with suffix 1, and the initial and final media with suffixes 0 and 2, 
the paths in the intermediate media not being explicitly considered. 
§ 3. Singularities and Convergence. 
Consider any symmetrical optical system, of which PL and QM (fig. 2) are the 
initial and final refracting surfaces. Let F 0 be the front focus of the system and 
UF 0 Y the caustic for backward-travelling rays which are parallel in the final medium. 
Fig. 2. 
This caustic, as is well known, wall usually be of the type shown in fig. 2, 
approximating to a semi-cubical parabola with a cusp at F 0 , and, to fix ideas, we shall 
suppose the point of the cusp to be turned to the left. In the opposite case, an 
obvious modification of the argument will be found to lead to similar conclusions. 
Any ray in the initial medium, which touches this caustic, must emerge parallel to 
the axis after passing through the system. 
Let I 0 be an object point on the axis behind F u and sufficiently near to it for a real 
tangent to be drawn from I 0 to the caustic and yet go through the system. This 
