Accuracy of the Internally Focussing Telescope. 891 



for the staff is m. Since the image is inverted m is 

 negative. Whatever the separation o£ the lenses may be 

 F and F' are conjugate points, since both are foci corre- 

 sponding to a parallel beam of light between the two lenses, 

 and the magnification associated with them is —fc/tc'. 

 Thus the difference between the magnifications at F' and S' 

 is m-\-/c/fc', and the 'actual separation of the points being x, 

 the relation between these quantities is 



xK — m + KJ/c' (2) 



From the ordinary relation between the separations of 

 corresponding pairs of object and image planes 



x-Y dniKJK,' = 0, (3) 



or alternatively from the relation for the object space 

 analogous to (2) 



v J -dK = ljm + K'/fc (4) 



together with (1) and (2), it follows on eliminating x and K 

 •that 



f/ --w' • * (5 > 



where 



M= ± 4/{A a -4*(*+..mtf / )}- ... (6) 



A rather subtle point arises in connexion with this ambiguity 

 of sign. It might be thought that A and M must always 

 carry the same sign since d should be zero when /c-j-7n/c' 

 vanishes. This argument is valid when k is positive, but if k' 

 is negative it is not correct since x must become infinite for 

 some position of the staff bet wet n S and F. The correct 

 conclusion is reached by noting that when m increases 

 numerically, that is when the staff approaches the telescope, 

 the power of the combined system must be increased to 

 keep the position of the image stationary. Now d, K, and 

 k are positive ; m is negative. Thus from (3) xk! is positive, 

 and further from (1) A is positive and greater than K. 

 Moreover 



K = i(A + M) (7) 



and therefore M must increase numerically with m. It 

 follows that M must have the same sign as k . 



In practice m, though never zero, is always small, but 

 this does not justify a solution of the problem by ex- 

 panding M binomially in a series of ascending powers of m 

 and retaining only the first two terms. The errors resulting 

 from such treatment would not be inappreciable, and would 



