276 ON THE GENERAL LAWS OF OPTICAL INSTRUMENTS. 



PROP. IV. Given the principal foci and principal planes of an instrument, 

 to find the relations of the foci of the incident and emergent pencils. 



Let F lt F t (fig. 3) be the principal foci, (?,, (?, the principal planes, Q t 

 the focus of incident light, <?,P, perpendicular to the axis. 



Through Q l draw the ray Q&F^ Since this ray passes through F l it 

 emerges parallel to the axis, and at a distance from it equal to G&. Its 

 direction after emergence is therefore Q t g t where G t g t = G^. Through Q l draw 

 P,y, parallel to the axis. The corresponding emergent ray will pass through 

 F tt and will cut the second principal plane at a distance G t y t =G l y 1 , so that 

 F t y, is the direction of this ray after emergence. 



Since both rays pass through the focus of the emergent pencil, Q t , the 

 point of intersection, is that focus. Draw Qf t perpendicular to the axis. 

 Then P& = G l y l = G t y t , and <?,, = ,#, = P,&. By similar triangles FJP& and 



And by similar triangles F,P 3 Q t and F t G t y t 



P&( = G^ **& ::GJP,: F,P,. 

 We may put these relations into the concise form 



PA _ P.Q. 



and the values of FJP t and P t Q t are 



, p Q _ 



/' A' *~ P f '> 



These expressions give the distance of the image from F t measured along the 

 axis, and also the perpendicular distance from the axis, so that they serve to 

 determine completely the position of the image of any point, when the princi- 

 pal foci and principal planes are known. 



PROP. V. To find the focus of emergent rays, when the instrument is a 

 telescope. 



Let Q l (fig. 4) be the focus of incident rays, and let <2,a,&, be a ray 

 parallel to the axis ; then, since the instrument is telescopic, the emergent 

 ray Q t a t b t will be parallel to the axis, and Q t P t = l. (?,P,. 



