30 Dr. Young’s Lecture 
Scholium 2. In all these cases, if the rays converge, d must 
be negative. For instance, to find the joint focus of two con- 
vex, or concave lenses, the expression becomes, e == 
Corollary 9. In Cor. 3, the divisor becomes ultimately con- 
stant; and, when the inclination is small, the focus varies as uu. 
Corollary 10. For parallel rays falling obliquely on a double 
convex, or double concave lens, of inconsiderable thickness, the 
radius being 1, e = 2 ; which varies ultimately as the 
product of the cosines, or as r ~~ t -f- C. 
Scholium 3. In the double convex lens, the thickness dimi- 
nishes the elfect of the obliquity near the axis ; in the double 
concave, it increases it. 
Scholium 4. No spherical surface, excepting one particular 
case, (Wood, 155,) can collect an oblique pencil of rays, even 
to a physical point. The oblique rays which we have hitherto 
considered, are only such as lie in that section of the pencil 
which is made by a plane passing through the centre and the 
radiant point. They continue in this plane, notwithstanding the 
refraction, and therefore will not meet the rays of the collateral 
sections, till they arrive at the axis. The remark was made by 
Sir Isaac Newton, and extended by Dr. Smith, (Smith r. 
4,93, 494.;) it appears, however, to have been too little noticed, 
(Wood, 362.) The geometrical focus thus becomes a line, a 
circle, an oval, or other figure, according to the form of the 
pencil, the nature of the surface, and the place of the plane re- 
ceiving the image. Some of the varieties of the focal image of 
a cylindrical pencil obliquely refracted are shown in Plate VI, 
Fig. 28. 
