Page:Elementary Text-book of Physics (Anthony, 1897).djvu/429

§ 337] is a double-convex lens. The $$r$$ of equation (111) is negative because measured from the lens away from the source of light. The second term of the formula has therefore a negative value, and $$p'$$ is negative except when $$\frac{1}{p} > (\mu - 1) \left( \frac{1}{r} - \frac{1}{r'} \right).$$ If $$p = \infty,$$ we have $$\frac{1}{p} = 0$$ and $$\frac{1}{p'} = (\mu - 1) \left( \frac{1}{r} - \frac{1}{r'} \right),$$ a negative quantity because $$r$$ is negative. $$p'$$ is then the distance of the principal focus from the lens, and is called the focal length of the lens. The focal length is usually designated by the symbol $$f.$$ Its negative value shows that the principal focus is on the side of the lens opposite the source of light. This focus is real, because the light passes through it. Equation (111) is a little more simple in application if, instead of making the algebraic signs of the quantities depend on the direction of measurement, they are made to depend on the form of the surfaces and the character of the foci. If we assume that radii are positive when the surfaces are convex, and that focal distances are positive when foci are real, the signs of $$p'$$ and $$r$$ in that equation must be changed, since in the investigation $$p'$$ is the distance of a virtual focus, and $$r$$ the radius of a concave surface. The formula then becomes To apply this formula to a double-concave lens, $$r$$ and $$r'$$ are both negative; $$p'$$ is then negative for all positive values of $$p.$$ That is, concave lenses have only virtual foci. For a plano-convex lens (Fig. 104, 2), if light be incident on the plane surface, $$r = \infty$$ and $$\frac{1}{p'} = (\mu - 1) \frac{1}{r'} + \frac{1}{p} \cdot$$ This gives positive values of $$p'$$ and real foci for all values of $$\frac{1}{p} < (\mu - 1) \frac{1}{r'}\cdot$$

For a concavo-convex lens (Fig. 104, 6) the second member of the equation will be negative, since the radius of the concave surface is negative and less numerically than that of the convex