Internally Focussing Telescope in Taclieometry. 

 where 



= 



893 



*'(a-*)(P-Q) 



B 



2*(P+Q)(aP-h6Q)' 



«»-& 2 <?)(P-Q) 



2<a + 6)(P + Q)(aP + 6Q)' 

 = «*(« + &) ■ 2'*' 



«(aP + 6Q) P *(P4-Q)^ 



c = 4^(a + />) 2 -{(a4-6) A -aP 5Q} i 



1/c 5 



±K 2 (a + b)(a? + bQ) 

 (A-P)(A-Q) 



+ 7 



7- 



(9) 



2«*(P + Q) 



It is at once evident that BH-/3 and B — ft, and hence B 

 are positive. That C is positive follows at once from its 

 definition in (8). Distances will be obtained from the 

 instrument on the assumption that in (8) may in general 

 be given its mean value zero. Thus the quantity known to 

 the surveyor as the constant of the instrument is the distance 

 from the vertical axis of the instrument to F minus B. 

 C divided by the multiplier for the staff readings, usually 

 100, is the correct separation for the stadia lines. (3 and 7 

 are lengths which specify minimum errors, and it will be 

 apparent in a moment that both have the same sign as k'. 



At the two end points for which M==P and M = Q, 

 6 has the value 4-1, and takes the other extreme value —1 

 for the intermediate point where M= (a~P + bQ)/(a + b). 

 Obviously accuracy is lost if either a or b be made zero, 

 since the error does not then rise to a stationary point 

 and subsequently fall. The character of the error, which 

 is determined by the ratio of j3 to 7, will usually be specified, 

 and two special cases call for particular consideration. In 

 the one case the magnitude ot the possible errors is made 

 independent of the distance measured by assigning the value 

 zero toy; this is the case usually considered, and is secured 

 by giving a and b the values V — q and \/'—p respectively. 

 A better choice both practically and theoretically is to 

 relate the magnitude of the possible error directly to the 

 distance under measurement. In this case /3 must vanish 

 and a and b are equal. 



Before proceeding further it will be verified that in these 

 two cases /5 and 7 agree in sign with re' . For when 7 = 0, 

 if p i s t ne magnification for the far distance and a and b 

 have the positive values of the square root, a > b. On the 

 other hand, since the power is less for the far point than For 



Phil. Mag. S. 6. Vol. 41. No. 246. June 1921. 3 N 



