Internally Focussing Telescope in Tachometry. 895 



two ends of the telescope requires f to fall within limits 

 somewhat closer than 



-i<n<h .' (ii) 



hat it must not be forgotten that if the lens is moved 

 by means of a rack and pinion or similar device the range 

 of movement is restricted to one-half of that implied by 

 these limitations. Substitution from (10) in (9) yields 

 results which are highly suggestive of the principles which 

 should be followed to obtain high accuracy in the estimated 

 distances; but these will not be detailed here, as a method 

 as described later by which a telescope can be designed to 

 give any required degree of accuracy. 



The simplest method of applying (9) to any given system 

 is to express M in terms of d, k, k\ and / or A. The formula 

 required is found at once by squaring (6) and eliminating m 

 hj (5), with the result 



M =- -fc'/fcd+s/f^A + fc'/Kd) 2 -^}, . . (12) 



where the square root agrees in sign with tc' . It should be 

 noted that the values of a and b need not be found. For if 

 L is the value of M for which #= — 1, 



B + /3 = «7*L | 



and } . . (13) 



4*2(0-7) = 2A-L-(A 2 -4* 2 )/L. J 



Instead of fixing L independently, it will be determined in 

 the two cases here considered by assuming in turn /3 = 

 rand 7 — 0. In the former case B is known from P and Q, 

 which are found by giving d in (12) the extreme values f 

 and n for the far and near points. Thus L and hence C — 7 

 .are known. C + 7 is found from the values of P and Q. 

 When 7 = 0, L is given by the solution of (13) : 



L = A-2,c\C-v) + 2K x /{l-A(G-r / ) + K 2 (C-r / y}, (14) 



where the square root has the positive sign, and the value 

 -of determined from / and n is used. 



Perhaps the clearest insight into the meaning of C is 

 gained by noting that 6 becomes unity at each end point, 

 and therefore the power for both these distances satisfies 



d= -B-+J9 + (C + 7) (dK + *'/*) . . . (15) 



by (4) and (8), and hence, if the powers corresponding 



3 N 2 



