MKMOIUS OF THE NATIONAL A«'AI)K.MY OF SllFNCFS. 45 



object. This process is applicable tn some forms of eoiiipotiiid luicroseopi' — sueli as thai \vlii<li is 

 used as a terrestrial ocular in tiie telcscojie — but is very exceptional. In practically every other 

 couvcr^in}j system the nef^ative jirincipal points will be outside the system and on opposide sides. 

 Find these and the two i)rineipal focal points, IIumi. as w»^ see from (</') and (»/") tin- principal focus 

 is exactly halfway between the principal point and the corres])onilin;,' ne;,'ative jiriiicipal point. 



'I he method of lindiu}: the four nodal points is j)recisely similar, except that the ma^rnilications 

 are taken as u.. and — />„ instead of 1 and — 1, as in the case of the four prlmipal points. 



VI. — MACNirifi.VTION (»!•• OPTK^AL SVSTKMS. 



The general expression for the magnification is equation (c), which can, ot course, be nioditied 

 so as to be jjiven in terms of the distance of the object from .r,„ or from the lirst vertex of the 

 .system, or indeed from any other i)oint fixed with respect to the system. Hut it is not easy to see 

 that this would be of any practical interest. A photographer may desire a tleliinte ratio of the 

 image in his camera to the size of the oI)ject, and no (hmbt he could tell at once how far theobjei-t 

 mu.st be from the camera to give this ratio, if he had (b'termincd the vabu^ of /.• for two determinate 

 positions of Xu and x,; but Ids method in practice of moving the instrument with respect to the 

 object until the image becomes of the desired size would involve no more measurements tlian that 

 of a single distance, which would al.so be necessary in the more recondite method. 



F<yr instruments used as aids to vision, however, the expie.ssiou for magnification becimies 

 l)articularly interesting, or rather the ex{)res.sion for augalar magnification, since we care nothing 

 for the absolute size of any image in question. 



^ d_ 



^ J3. 



Fig 4 



Let Fig. 4 represent any optical apparatus to be used as an aid in seeing 0'. Let n" be an object 

 m such a place that its image is at rf", and very lu-ar the place of the ej-e. Call the distance from 

 the eye to the object />, and the distance from a" to the object d, as represented in the figure. If 

 d is very large, the in.strumeiit is called a telescope; if small, a microscope; when neither one nor 

 the other, we have no name for it. For example, the instruments emiiloyed as optical aids in read- 

 ing the distant circles of an equatorial may be called with equal propriety telescopes or micro- 

 scopes. In short, there is no precise distiiu-tion between the two types and a general e(iuation of 

 the niagnifjing power otiglit to be api)licable as well to one as the other. 



We will define the niagnifying ])ower of such an instrument as the ratio of the apparent 

 dimensions of a small object in the axis, as seen through the instrument, to that of the same ob- 

 ject seen without the instrument. 



From the diagram we see tli; 



is also twice the angle cf>„ for the point of the object farthest from the axis. The angular sub- 

 tense of the object, as seen "from «," through the instrument is 2(p„ while the value of this angle 

 without the instrnment is ^^, as is evident from the figure. This angle we will call 2<P. The mag- 

 nifying power is therefore eciual to 



