I 



ON THE THEORY OF OPTICS. 327 



\yiien a radiant point is at twice the distance of the principal focus 

 ' from fc convex lens, the image is at an equal distance on the other side ; 

 when the radiant point is nearer than this, the image is more remote, the 

 distance of the image from the principal focus nearest to it being always 

 inversely as the distance of the object from the principal focus on the 

 opposite side. (Plate XXVII. Fig. 385.) 



The joint focus of two lenses, in contact with each other, is also found by 

 multiplying together their separate focal lengths, and dividing the product 

 by their sum or difference, accordingly as they agree or differ with respect 

 to convexity and concavity. 



We have hitherto considered the place of the focus only in relation to a 

 single point, placed in the axis of the lens or mirror ; but it is equally 

 necessary to attend to other points, out of the principal axis ; for in order 

 to form a picture, the rays from a great number of such points must be 

 collected into as many distinct points of the image. Some of the rays 

 proceeding from every radiant point must be considerably bent, in order to 

 be collected into a common focus ; others remain nearly straight ; and if 

 we can discover which of the rays are ultimately either in the same line 

 with their original direction, or in a direction parallel to it, we may 

 determine the line in which the image of the point in question is to be 

 found. For this purpose we employ the property of the optical centre, 

 which is a point so situated, that all rays which pass through it, or tend 

 towards it, while they are within the lens, must ultimately acquire a direc- 

 tion parallel to their original direction. In some cases, the optical centre 

 may be without the lens, but no practical inconvenience results from 

 supposing it to be always situated within the lens, especially when its 

 thickness is inconsiderable ; so that all rays which pass through the middle 

 point of the lens must proceed, without sensible error, in the same straight 

 line, and the image of any radiant point must consequently be found some- 

 where in this line : but in the case of a mirror, the centre of its figure is 

 also the optical centre. Now when any radiant point is removed a little 

 from the axis of a lens or mirror, the distance of its image is in general a 

 little diminished, but the difference is too small to be observable in common 

 cases. We may, therefore, suppose it to be at the same distance as if the 

 point remained in the axis, or even to be in a plane crossing the axis 

 perpendicularly at that distance, so as to form part of a flat image, of 

 which the magnitude is determined by straight lines drawn from the ex- 

 tremities of the object through the centre of the lens. This is, however, an 

 approximation which is only admitted for the greater convenience of com- 

 putation and representation, the image being almost always in reality 

 considerably curved. (Plate XXVII. Fig. 386.) 



LECT. XXXV. ADDITIONAL AUTHORITIES. 



, Optics in general. Euclidis Optica, 4to, Paris, 1557. Faulhaber, Descriptio 

 Inst. Geom. et Opt. 4to, Frankf. 1610. Kepler, Dioptrice, 4to, Augsb. 1611. 

 Aquilo, Op. Autw, 1613. Schneider de Luce, 1616. Mathise Buchholdii Lucis 

 Contemp. 4 to, 1630. Descartes, Dioptrique, 1637. Bwllialdus de Natura Lucis, 



