392 TUB RELATION OF GEOMETRICAL OPTICS TO MATHEMATICS AND PHYSICS. 



where r,, r, are measured from the instrument in opposite directions along the 

 axis of the pencil in the media /*,, j*,, respectively, and x,, o^ are perpendicular 

 to the axis. 



a,, a, are the values of r,, r t , for the principal foci, and / / the prin- 

 cipal focal lengths, and /,/t, = 



If V 5 -^-!?' 



/I l i ^t 



the last term of V assumes the form - , and an infinite number of possible 



paths exist between the points (,, r,), and (a^, r,), which are therefore con- 

 jugate foci. 



Differentiating V with respect to a^ and x t we obtain 



1 



Z> is the quantity in Cotes' Theorems which Dr Smith calls the Apparent 

 Distance, or the distance at which the object must be placed that it may 

 subtend the same angle as when viewed through the instrument. 



We have also 



dD . dD 



