OF OPTICAL INSTRUMENTS. 



77 



tance ; and when it is within the principal 

 focus, the magnified image is virtual, and 

 erect : but a concave lens and a convex 

 mirror, always form a virtual image, which 

 is erect, and smaller than the object. 



For in a lens, if the rays con- 

 verge after refraction, it must 

 be to a point beyond the centre, 

 and the rectilinear rays will de- 

 cussate in the centre ; and if 

 hey diverge, it must be from a 

 point on the same side of the 

 centre with the object, and the 

 rectilinear rays have not cross- 

 ed. In the concave mirror 

 the foci are always on opposite 

 sides of the centre of the sphere, 

 since the sum of the reciprocals 

 of their distance is equal to 

 twice the reciprocal of the ra- 

 dius (418), except when the 

 object is within the principal 

 focus, and then there is an 

 erect virtual image beyond the 

 Surface. In the convex mirror 

 the image is always virtual and erect, being between the 

 surface and the principal focus (419) ; and m the plane 

 mirror the image is obviously erect andeijual to the object. 



439. Theorem. The image of any ob- 

 ject formed by a spherical reflecting surface 

 subtends the same angle as the object both 

 from the surface and from its centre. 



It is obvious that the rays which^ass through the centre 

 must remain in the same right line ; and since in this case 



aa — ad 



the distances from the centre are a— d and e—a, and (/and 

 e are the distances from the surface ; consequently the 

 image and object are in both cases the bases of similar tri- 

 angles. 



440. Axiom. The intensity of light is 

 inversely as the surface on which any given 

 portion of it is spread. 



Scholium. Hence the illumination is said to decrease 

 as the square of the distance increases. 



441. Theorem. The illumination of the 

 image, formed by any lens or mirror, is equal 

 to that which would be produced by the im- 

 mediate effect of the surface of the lens or 

 mirror, if equally illuminated with the object. 



Supposing the whole quantity of light that falls on the lens 

 or mirror to be collected into the image, the condensation 

 is in the ratio of the surface of the lens to that of the image. 

 Now the illumination produced by a surface equal to the 

 image at the distance of the lens or mirror, is equal to the 

 illumination produced by the object at its actual distance, 

 supposing the brightness equal, since the linear magnitudes 

 of the object and image are proportional to their distances 

 from the lens or mirror, and the surfaces are as the squares 

 of the distances ; the intensity _of the light falling on the 

 lens is therefore such as the supposed surface would pro- 

 duce; and when this is increased in the ratio of the surface 

 of the lens to that of the image, it becomes equal to the il- 

 lumination produced by the surface of the lens, supposing 

 it similar to that of the luminous objiect. 



442. Theorem. The intensity of illu- 

 mination of the image of a luminous point, 

 formed by a spherical surface, is inversely as 

 the fourth power of the cube root of the dis- 

 tance from the centre. 



The quantity of light which falls on any portion of the 

 surface is as the square of its sine xx, or as the versed sine 

 y i and the lateral aberration varies as the longitudinal 

 aberration and as the aperture conjointly, that is as xy o» 

 as x' ; now the intensity of light is as the fluxion of the 

 quantity of light, divided by the fluxion of the surface, or as 

 2xi 1 



J^> or as—, or inversely as the fourth power of the aper- 

 ture, or of the cube root of the radius of the circle of aber- 

 ration. 



Scholium. This is not the least circle of aberration 

 but it is probably the circle in which the aberration has the 

 least effect in producing indistinctness, and therefore it must 

 be considered as determining the degree of distinctness bi 

 the image. 



443. Theorem. If the whole of the 

 light falling uniformly on an infinitely small 

 sphere were regularly reflected, it would be 

 scattered equally in all directions. 



The quantity of parallel rays falling on a ring, of which 

 the breadth is z', the evanescent increment of the circle, and 

 represented by a hollow^ cylinder, must be as xx', x being 



