FIEST ORDINARY MEETING. 17 



tion, and that the figure also gives the relation /\ A'= /' ^^^^ "> 

 where A' = P"<^- 



In the case of a convex mirror X and Y will lie in the opposite 

 quadrant and the longitudinal aberration will be found to be 



VA dj' 



III. 



28. Since writing the above it has occurred to me that the relation 

 dd' =//' leads to two other simple geometrical methods for exhibit- 

 ing the relations between the conjugate points. 



Thus if we separate the two axes FF', FF' so that F in the x axis 

 coincides with F' in the y axis, as in Fig. 14, then evidently the feet 

 of the ordinates drawn from any point on the hyperbola xy ^=ff' 

 will be conjugate to one another. This construction gives us a 

 readier means of finding many of the points whose positions have 

 already been discussed. 



Thus self-conjugate points are at once given by 

 X {2h — x)—ff' ; 

 and the points K, K' (§ 15) by 



2hx =//'. 



Again, H being the middle point of FF', if H is the image of Gr, 

 and J of H, we have 



F'J == ^ = 2FK = FG. 



h 



29. From the consti-uction of the preceding section it appears that 

 the lines joining pairs of conjugate points on the two axes touch the 

 hyperbola 



Fig. 14 shows that the conjugate points V, Y' are equidistant 

 from H, the middle point of FF', and that 



FY = F'Y' = FT = yff\ 



Prof. Galbraith, Mr. Wm. Houston, and Mr. A. Baker took 

 part in the discussion which followed. 



2 



