4.18 LECTURE XXXV. 



radiant point is nearer than this, the image is more remote, tlie 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 neces- 

 sary 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 col- 

 lected into as many distinct points of the image. Some of the rays proceed- 

 ing from every radiant point must be considerably bent, in order to be col- 

 lected into a, common focus; others remain nearly straight; and if Ave 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 direction 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 irnage of any radiant point must 

 consequently be found somewhere 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 ob- 

 servable 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 



