1897.] near the Focus of a Telescope. 265 



which as in the preceding case may be reduced to 



W,- ^ = -1^(2)+^^- P -Q-i(P -Q)\ e ^, 



giving 



0i = ~f K 1(1 -P-Q) ^n fut -(P - Q) cos pg\ , 



U 2 = - J (z) - £~ 1(1 - P- Q) cos p« + (P - Q) sin /^j , 



and mi 2 = U 2 + U.r 



=i{Q--P-Qy + (P-Qft 



-U (*) {(1 -P-Q) cos fiz + (P - Q)sin fiz} 



or If 2 = J/ 2 2 + e', 



where 8 2 l/ 2 2 = £ {(1 - P - Qf + (P - Q) 2 } 



SV = - | Jo (*) ((1 -P-Q) cos p« + (P - Q) sin ^J. 



The values of if, 2 and il/ 2 2 may be calculated from Gilbert's 

 tables for P and Q : it will be remembered that the argument in 

 this case is 



V 



2 (/A -1)3 



7T 



-*) 2 g 

 7T/C 



3> \ /3/ 



= ± 1- 



the upper sign being taken in ilfj 2 and the lower in ilf 2 2 , so that v 

 is proportional to the distance of the point of the image considered 

 from the geometrical boundary. The curve in the figure represents 

 h 2 M? and h 2 M 2 for different values of the abscissa v, the origin 

 corresponding to a point on the geometrical boundary and 



u. 



IT 



to the centre of the image. The numerical values of PM 2 and 

 &M.? are given in Tables I. and II. The figure shows that the 

 intensity increases from zero outside the boundary until it reaches 



