44 MEMOIRS OF THE NATIONAL ACADEMY OF SCIENCES. 



Oase 2. — Should we desire to find the constants of [d) by experiment, proceed as follows: 

 Choose any convenient point in the axis of the system for x^, placing there an object, e. g., a 

 scale of equal parts; the image of this object will be at Xi and the ratio of the size of the object to 

 the image will be Ic. Then find the place on the axis of the image of an object at an indefinitely 

 great distance in front of the system, that is when G'=0. The reciprocal of the distance of this 

 point from Xi equals Icncfx+i, and the problem is solved. 



V. — ON PAETICIILAR VALUES OP k IN EQUATION {(l). 



Inspection of the equation (d) suggests at least four values of k, which make the equation of 



special simplicity. These values are 1, —1, — and . Substituting these in turn we have 



Po — Po 



Cp =ae\+i + noCP (di) 



G_p=—ac\+x-\-iJ>G-p (d") 



The points defined by .ro and Xi when A:=l are called the first and second principal points, 

 respectively. They were first introduced and their properties defined by Gauss, in a celebrated 

 paper published in 1840. We see at once that the image of a small object at Xa is at aii, the trans- 

 verse magnification is 1, and the longitudinal magnification is, from (/), — . The inclination of 



the incident wave at j-q and the finally refracted wave at Xi is given by {g) which becomes 



sin yi _ 



sin <po ~ 

 For /i-= — 1, the second of the above equations, the points Xo and Xi are called the first and 

 rsecond negative principal points. An object at x„ has its image at .ri, the image being inverted, 

 but of the same transverse dimensions as the object; the longitudinal magnification is the same 



as before, namely, — . Finally (5") gives 



sin qji _ 

 siu^o""'^'" 

 For A:= — , the two points x^ and x^ are called the nodal points. These were first investigated 



po 



and named by Listing in 18.51. An object at the first nodal point has its image at the second 

 nodal pointi both axial and transverse magnifications being equal to po- These are the only two 

 points so related that the image of a body at one point has the same shape and orientation as the 

 body itself. A more important property is derived from equation (</), in which we see for this case 



sin q)i_ 

 sin 9>o 



tliat is, a wave surfiice which would pass through the first nodal point at an inclination <^„ passes 

 through the second nodal point under the same inclination after final refraction. 



The final form, in which the points Xa and X\ may be called the negative nodal points, has 

 the same longitudinal magnification for these points, but the transverse magnification and the 

 relation of the inclinations are equal to those of (rf'") taken negatively. 



It will be observed that (fZ') is of exactly the same form as the equations for a system of 

 infinitely thin lenses in contact. Moreover, if the first and last mediums are alike — in which case 

 ,j„=l — equations (d>) and [d™) become identical, as do also (fZ") and {d'"), or, in words, the prin- 

 cipal and nodal points fall together, and also the negative principal and negative nodal points. 



The determination of the position of all these points is the problem, when the constants of 

 ihe system are known, of Case 1 of the preceding section, and therefore need not be further 

 discussed. But to deteimine them experimentally is not the same as Case 2, because they 

 assume determiuiite values for k. We may proceed as follows: 



If both princi]ial ixdnts are (mtside of the system and on opposite sides we may find them at 

 once by seeking the places of object and image when the image is erect and ecjual in size to the 



=1 



