390 HAMILTON'S CHARACTERISTIC FUNCTION FOB A NABROW BEAM OF LIGHT. 



< 38 > ; 



or. in woixls, The distance of the, object from the axis is to the distance of 

 the uiHiyc from the axis as the distance of the object from the first principal 

 focus is to the first principal focal length, or as the second principal focal 

 length is to the distance of the image from the second principal focus. 



Let A, be the distance at which an object of diameter x, must be placed 

 from the eye that it may subtend an angle equal to that which it subtends 

 when placed at z,, and seen through the instrument by an eye at z,, 



/r, = TT when x t = Q, or A, = ' fU ................... (39), 



or 



The quantity /&, is that which occurs in Cotes' Theorem, and to which 

 Smith gives the name of the "apparent distance." 



Differentiating h t with respect to z l and z,, we find 



dh_ l _z,-u ? dh l _z 1 - u, d'h, 1 , 



~ - ' -- 



When the focal length is infinite, the instrument becomes a telescope, and the 

 characteristic function is 



IT- IT- . , . 1 iu,LLJ lt . ., 



7= F. + ^z. + f^j + l - r -^r 1 + a sim Jar term in y. 



--*- 



Here m is the angular magnification, and u lt w, are the co-ordinates of any 



two conjugate foci. The linear magnification is -^- and the elongation is -~ 



- 



