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1. The plano-convex, Fig. 81, which may be considered as a variety of this, the radius of one of the spheres becoming infinite.
2. The double-concave, Fig. 82.
3. The plano-concave, Fig. 83.
4. The meniscus, Fig. 84, bounded by a concave and a convex surface which meet.
5. The concavo-convex, Fig. 85, in which the surfaces do not meet.

86. Prop. To find the direction of a ray after refraction through a lens.

The method we shall follow here is to consider a ray refracted at the first surface, as incident on the second, and there again refracted; we shall have occasion to add to the letters hitherto used

∆″ for the distance of the focus after the second refraction,
t the thickness of the lens;
r′ the radius of the second surface.

Then taking, for instance, the concavo-convex lens in which both the centres are on the same side, (Fig. 86.)

1/∆′=1/m∆+m−1/mr for the first refraction,

1/∆″+t=m/∆′+t−m−1/r′, for the second,

t being added to ∆′ and ∆″ as the distances are now to be measured from the second surface. However, in order to simplify the expressions, it is usual to suppose the thickness of the lens inconsiderable in comparison of ∆′ and ∆″, in which case we may write

1/∆″	=	m/∆′−m−1/r′
	=	1/∆+m−1/r−m−1/r′
	=	m−1 (———)(1/r−1/r′)+1/∆.