In the combination of lenses with the flame of a lamp, similar considerations must influence us in making the necessary arrangements, as in the case of reflectors. We have already seen that the size of the flame and its distance from the surface of reflecting instruments have an important practical bearing on the utility of the instrument, and that the divergence of the resultant beam materially affects its fitness for the purpose of a Lighthouse. So also, in the case of the lens, unless the diameter of the flame of the lamp has to the focal distance of the instrument a relation such as may cause an appreciable divergence of the rays refracted through it, it could not be usefully applied to a Lighthouse; for, without this, the light would be in sight during so short a time, that the seaman would have much difficulty in observing it. To determine the amount of this divergence of the refracted beam, therefore, is a matter of great practical importance, and I shall briefly point out the conditions which regulate its amount, as they are nearly identical with those which determine the divergence of a paraboloïdal mirror illuminated by a lamp in its focus. The divergence, in the case of lenses, may be described as _the angle which the flame subtends at the principal focus of the lens_, the maximum of which, produced at the vertex of FRESNEL’S great lens by the lamp of four concentric wicks, is about 5° 9′.[60] [60] This will be easily seen by examining the annexed figure (64), in which Q _q_ represents the lens. A its centre, F the principal focus, _b_ F and _b′_ F the radius of the flame; then is the angle _b_ A _b′_ equal to the maximum divergence of the lens. _b_ F Rad. of flame Sin _b_ AF = ----- = sin _b′_ AF = --------------; AF Focal distance and twice _b_ AF = the whole divergence at A. Then for the divergence at the margin of the lens, or at any other point, we have, FQ = √(AQ² + AF²) and Q _x_ = √(QF² + F _x_²); and for any angle at Q, we have F _x_ sin FQ _x_ = -----. FQ [Illustration: Fig. 64.] ~Illuminating Power of Lenses.~