330 



Mr. R. F. Muirhead on 



Next let P' (Rg. 5) be the image o£ anv point P. Draw 

 PU' parallel to the axis to meet P' Fo' in U'. Draw P' V 

 parallel to the axis to meet P Fi in V. Let the perpendi- 

 culars from Y, U' on the axis meet the axis in H, H' and 

 PU', P T in U, y. 



Fig. 6. 



p\ 



1 



Fig. 5. 





U 



\.f? 





H 









W 





2 





z' 







X 







Y'\ 







Y 



\ 





^ 



\^ 





F, 



H 



H 



h' 



' 



f; 



p' 



Since the in-ray P F^ Y has Y Y'P' for out-ray, it follows 

 that the image of Y is in YP^, and hence that the vertex for 

 H is at infinity. Hence the image of U is in PUU'. But 

 it also lies in U' Fo' P'. Hence U' is the image of U. Hence 

 H' is the image of H. 



The vertex of H being at infinity^ the raaanincation for the 

 plane of H is unity. Hence H and H' are the first and 

 second principal points. 



From W (fig. 6), a point in the first focal plane, draw 

 \^ 7.7.' parallel to the axis to meet the planes of H and H' 

 in Z and Z'. Z' is the image of Z, and Z' F^' is the out-ray 

 for W Z, since W Z is parallel to the axis. Draw W X 

 parallel to 7' Fo' to meet the axis in X. Since "\Y is on the 

 focal plane, all in-rays through it emerge parallel to one 

 another, and therefore to Z' Fo^ Hence the out-ray for W X 

 is parallel to it. 



To find the position of its out-ray let W X cut the plane of 

 H in Y, and draw Y Y' parallel to the axis to meet the plane 

 of H' in Y', and draw Y'ls' parallel to WX to meet the axis 

 in X'. Then Y' is the image of Y. and therefore the out- 

 rav for TTX passes through Y', and is in the line X' Y^ 



Since FX = H'F' and HX = H'X' it follows that the 

 positions of X and X' are independent of that of TT, so long 

 as W is in the first focal plane. 



Hence all in-rays through X have their out-rays through 

 X' and unaltered in direction. Thus X and X' are the jirst 

 and second nodcd points. 



We have seen that when the axis of the system is given 

 the whole system becomes determinate when two extra-axial 

 object-points (not in the same normal plane) and their images 

 are given. The following data are also sufficient to enable 



