278 ON THE GENERAL LAWS OF OPTICAL INSTRUMENTS. 



tr/gr,' = 6,'gr,'. ^ is by definition the second principal focus of the combination 

 of instruments, and if r,y, be the second principal plane, then r,y J =(r I gr 1 . 



We have now to find the positions of <, and F,. 

 By Prop. IV., we have 



Or, the distance of the principal focus of the combination, from that of the 

 second instrument, is equal to the product of the focal lengths of the second 

 instrument, divided by the distance of the second principal focus of the first 

 instrument from the first of the second. From this we get 





Now, by the pairs of similar triangles tfG t 'g t ', <r a y, and FjGrfgf, F t G.g v 



IVy, %, _ F& 



Multiplying the two sides of the former equation respectively by the first and 

 last of these equal quantities, we get 



r j _ j, __* 



Or, the second focal distance of a combination is the product of the second 

 local lengths of its two components, divided by the distance of their consecutive 

 principal foci. 



If we call the focal distances of the first instrument y, and f t , those of 

 the second /,' and /,', and those of the combination / / and put 

 then the positions of the principal foci are found from the values 



d ' ' 

 and the focal lengths of the combination from 



