792 Transactions of the Society. 



Secondly, we have essential defects in the performance of ohjectives ; 

 the accumulated influence of certain slight imperfections in the 

 technical work of the lenses and certain residuary aberrations 

 which cannot be eliminated by the most skilful construction under 

 the actual conditions of optical work at the present time. These 

 alone can claim a general signification, and admit of an approximate 

 estimation according to the existing standard of optical art. 



In such an estimation we do not need any detailed analysis of 

 the various soui'ces of defective performance. For our present 

 purpose it is quite sufficient to enunciate certain optical proposi- 

 tions, by means of whicli the problem may be reduced to one 

 question, to be answered on the grounds of practical observation. 



It may be easily shown, on well-established principles, that 

 with one and the same objective the total effect of all essential 

 aberrations, if measured by the linear diameter of the dissipation- 

 circles in the image, always varies in direct ])roportion to the linear 

 amplification of that image, provided the distance to which the 

 image is projected is a considerable multiple (not less than about 

 the ten-fold) of the clear opening of the objective. This holds 

 good (with the restriction just named) for every position of the 

 image, and whether this is changed from a real image to a vhtual 

 image, and vice versa — that is to say, that if the linear amplifica- 

 tion is increased in the proportion of 1 : n by projecting the image 

 to a greater distance irom the objective, the dissipation-circles 

 which appear instead of sharp points are always increased in the 

 same proportion, if the accidental aberrations attendant upon the 

 change of the conjugate foci are eliminated. This latter condition 

 means that if an objective has its hest correction for a certain 

 distance (A) from the back of the objective, and the image is now 

 projected to another distance, n A, on the same side (or on the 

 opposite side, the image being virtual in the latter case), the cor- 

 rection will probably be largely deranged by the alteration, and a 

 large amount of new aberrations introduced thereby. But if this 

 is properly compensated by any of the ordinary means, and the 

 hest correction for the new position of the image is obtained, the 

 residuary aberration will be reduced to an amount which will 

 exactly coiTespond with the change in the amplification according 

 to the above rule.* 



This statement leads now to several inferences of practical 

 importance, which are : — 



(rt) The total efi'ect of the aberrations (therein including the 



* The tacit assumption which is implied in the proposition that the com- 

 pensation for cliange of tlie conjugate foci is always possible, without introducing 

 new aberrations and without altering the focal length and the aperture, may be 

 readily shown to be true under the restrictions in regard to the distance of the 

 images which liave been indicate^ above. 



