714 Mr. W. Bennett on 
there will be two rays only, one from each surface. Evidently, 
then, the virtual shadow passes through much the same forms 
as we have already seen for the real. 




If the virtual shadow were drawn in fig. 4 it would be 
nearly straight, very close to the axis, and on the same side 
of it as the object. 
Referring back to the Cartesian equation, we see that there 
are two asymptotes parallel to the axis of y given by 
a?(w—d)?—2(cd—ba)?=0 ; 
or ea ate? , | 
atby/2 q 
so that we have always two open branches to the shadow, one 
real and the other virtual, and a closed branch which may be 
real or virtual, which contracts to a point when the object 
touches the caustic surface and vanishes when it does not 
meet it at all. 
The Cartesian equation gives y? as a single-valued function 
of w and affords an alternative method for plotting the shadow 
forms. 
If the pencil is not parallel before reflexion, or if we take 
a pencil rendered convergent by a lens with spherical surfaces, 
we get in general a less simple expression for the longitudinal 
aberration, and the equation of the shadow takes a more com- 
plicated form. The real shadows are, however, of much the 
same appearance. Fig. 8 (Pl. XX VII.) shows a series of photo- 
graphs of shadows of a straight wire in a pencil produced by 
refraction at a plano-convex lens. The lens used was of about 
10cms. diameter and the radius of curvature of its spherical face 
was 7*8cms. The disposition of the apparatus was as follows :— 
An image of the filament of a Nernst lamp was focussed 
on to a card screen pierced by a pinhole, so that the image 
