The value of B b gives, of course, the direction of the second tangent Z b (which must be equal in length to AZ), whence we easily deduce the chord of the reflecting side A b.
[68] The following steps will shew the mode of obtaining this expression: Suppose ([fig. 74], on opposite page) F n to be a ray incident on the surface BC very near b or B (which, although exaggerated in the figure for more easy reference, are close together), and let this ray F n be refracted in the direction n O, and draw n n′ parallel to CA, the ray which is refracted at C, then will n′ n O = m . b FC = ²⁄₃ b FC. But the tangent AZ should make with the tangent b Z an angle equal ¹⁄₃ b FC, or one-half the inclination of the rays refracted at b and C, which are afterwards by the agency of those tangents, to be reflected in the directions parallel to b C and to each other. Hence we have AX b (which is the inclination of the normals to those tangents),
or AX b = BZ b = b FC 2 m = b FC 3 nearly.
Fig. 74.
But putting AXB ([fig. 73], p. 277) for AX b, and BFC for b FC, a supposition which may be safely made when the differences are so small, and founding upon the analogy
AX ∶ AB ∷ R ∶ tan AXB, we have BA = AX . tan AXB = AX . tan ¹⁄₃ BFC. Then
| AB² | = | B β (B β + 2 AX) |
| = | B β² + 2 B β . AX |
and neglecting B β², which is very small, we have:
BA² = B β . 2 AX nearly,