KEPETITION OF ANGULAR MEASURES WITH THEODOLITES. 95 



Each angle in the third column in I. is affected by two errors of 

 reading and two of pointing, see (17), but in the summation they 

 are all excepting the first and last, cancelled out of the result, see 

 (20), or (13). Thus although the angles are symmetrical in 

 respect of their errors, to find the arithmetical mean of n measures 

 is exactly equivalent to dividing the (w+l)th reading by n, and 

 the intermediate readings have no influence on the result, a result 

 liable to any abnormity in the two readings on which it depends, 

 and yet having a higher probability than n independent measures 

 liable to the same errors. 



Forming in I. the residuals, (and their squares) in the usual 

 way, by subtracting the mean 89° 59' 52'9" from each measure, 

 see v and v 2 , or preferably by subtracting 99° 59' 52-7", which 

 will hereafter be shewn to be a more probable value of the angle, 

 see v and v" 2 , the probable error of a single measure subject to 

 two errors of reading and two of pointing may be found. This, 

 employing (22) instead of (18), the desirability of which a general 

 knowledge of the magnitudinal relation of the errors of reading 

 and pointing suggested, and the final results sufficiently confirm, 

 is + 6*3", vide I. 



Proceeding similarly in II. but employing (18), the mean of ten 

 measures affected with twenty errors of pointing and two of read- 

 ing is found to have a probable error of + 0*6 6". 



In the example taken, the altitudes of the objects were both 



zero, but in view of the deductions expressed in (6) to (12) it will 



be desirable, in order to exhibit the form of the general solution, 



to put q — probable value of each 'pair of errors of pointing. Thus : 



q=*/(p' 2 +p" 2 )=p o v(sec 2 p x +sec 2 3 ) (24). 



Using the general notation of this article, the probable error of 

 a single measure in I. is + q as regards pointing, and + g o v2 in 

 respect of reading, the probable error of the sum of the two being 

 + v(q 2 + 2g o 2 ) = +6*3", and similarly the probable error of the mean 

 of ten " repeats " in II. is + VW'+ggo') = ± Q . 66 * whence, by 



multiplying both sides of the latter equation by 10, and then 

 squaring both equations, those hereunder are obtained , viz. : — 

 q 1 + 2g 2 = 39-69" and I0q 2 + 2g 2 = 43'56" ; from the solution 

 of which q 2 is found to be 0'43",°g to be 0-66", g 2 to be 19.6", 

 and g o to be 4-43".* 



Having found q and g o , the probable error of the mean results 

 of either I. or II. may be ascertained in the manner sketched 

 hereinafter. Thus the probable error of the latter is shown to be 



* The quantities are expressed to a high degree of precision not as 

 indicating their reliability to within such limits or that these limits 

 ought in general to be regarded as significant, but in order to more 

 clearly exhibit the processes. 



