894 Mr. T. Smith on the Accuracy of the 



the near, P<Q. The denominator is essentially positive, 

 and thus the sign agrees with that of: k' . When /3 = 0, 

 7 takes the form (P — Qi 2 /16V(P + Q) and all factors are 

 positive with the exception of P + Q which has the * ame 

 sign as k . 



As would be expected, C is approximately equal to the 

 focal length of the system when focussed for infinity, but 

 the stadia lines will lie incorrectly placed if their positions 

 are determined by the aid of collimators making with one 

 another an angle of one-hundredth of a radian. For if R is 

 the value of M for wi = 0, it is easy to throw the second 

 formula for into the form 



2 (P-R)(Q-R) 

 ^~A+R 2**(P + Q) T 



The two factors P — R and Q — R are positive, and thus 

 both the terms dependent on the range have the same sign. 

 If the focussing lens is positive, the stadia lines must be 

 closer together than the separation determined by the 

 collimator method ; if negative, the separation must be 

 greater. The most satisfactory method of spacing the 

 lines is to determine their correct distance by calculation. 

 For this purpose, somewhat greater accuracy than a slide- 

 rule gives is required. 



The significance of the expressions found for the constants 

 and errors is more readily appreciated by expressing them 

 in terms of other variables. It has hitherto been necessary 

 to regard M as a function of m defined by equation (6) to 

 enable d to be expressed in the form (8). This is no longer 

 necessary, and equation (7) shows that it may be regarded 

 as a power, and thus be expressed in terms of the powers of 

 the two lenses and of the position of the focussing lens in 

 the telescope. The equations required are 



A =2/C + K f -lKfc' 3 1 



M = «'(l + f«). | . . . . (10) 



where I is the constant sum of the separations of the 

 focussing lens from the stadia lines and from the objective, 

 and f is the excess of the former distance over the latter. 

 The separations are of course measured from the principal 

 planes, so that the length of the telescope will be rather 

 greater than / plus the space required for the eyepiece. 

 The second formula confirms the result, of which extensive 

 use has already been made, that M and k have the same 

 sign. The presence of the objective and graticule at the 



