Page:Elementary Text-book of Physics (Anthony, 1897).djvu/424

410 all other points of the mirror shall send light to the same point, or that the reflected wave shall be a sphere with its centre at $$l.$$ In order that this shall be the case, the distance $$LB + Bl$$ must be constant wherever the point $$B$$ is situated on the reflecting surface. Draw $$BD$$ perpendicular to the axis of the mirror. Represent $$BD$$ by $$y,$$ $$AD$$ by $$x,$$ $$LA$$ by $$p,$$ $$lA$$ by $$p',$$ and $$CA$$ by $$r.$$ Then we have $$LB = \sqrt{(p - x)^2 + y^2},$$ and $$y^2 = (2r - x)x = 2rx - x^2.$$ Hence follows If the aperture be small, $$x$$ will be small in comparison with the other quantities, and we may obtain the value of $$LB$$ to a near approximation by extracting the root of this expression and omitting terms containing the second and higher powers of $$x.$$ We obtain  In like manner we have  whence  When $$B$$ coincides with $$A,$$ the above value becomes $$p + p',$$ and the condition that all values of $$LB + lB$$ are equal, whatever be the value of $$x$$ within the limits already set to it, is found from  From this equation we obtain $$\frac{r}{p} + \frac{r}{p'} = 2$$ and $$p' = \frac{pr}{2p - r}\cdot$$ For the apertures for which the approxiniations by which the result was arrived at are admissible, the wave surface is practically spherical, and the point, the distance of which from the mirror is given by this equation, is the centre of the reflected wave. Since the disturbances propagated from $$bb$$ reach $$l$$ simultaneously, their effects are added, and the disturbance at $$l$$ is far greater than at any other point. The effect of the wave motion is concentrated at $$l,$$