98 G. H. KNIBBS. THE THEORY OF THE 



of the true angle and the real errors) of the multiple measures 



denned in column 1 of II., viz. : — 



10a + {-p x + p 2 - +P2o)~9i + ffn, 



10a + (~p 3 +p 4 - +P22)-g 2 + 9i2) 



10a + (-P19+P20- +P3s)-yxo+92o- 



and the addition of these is, — 



100a+{l(-p 1 +p 2 )+2( -p 3 +p^)+S(-p5+Pe) + 10( -p^+poo) 



+ 3 (-p 33 +p3*) + Z(-P3 6+Pse)+l(-pa7+p S s)} + {~9i-y<2 



- ~9io+9ii + +919+920) (26), the error of the 



mean result being one hundredth of the terms following 100a, 

 inasmuch as it is the one hundredth part of the whole quantity. 

 Putting q as before for the probable value of ( - p'+p"), the probable 

 value of the p term is that of: — q (1 + 2 + 3 + .. . + 10 + .. .. + 3 + 2 + 1) 

 and this is, by the theory of errors, g\/(l 2 +2 2 + 3 2 + ....10 2 + ....3 2 + 



2 2 + l 2 ) (28). Proceeding to numerical evaluation, # 2 =0-61", 



the sum of the series - 670, therefore v(670 x 0-61") = V408-7" = + 

 20*22" the probable error of the p term. That of the g term is 

 V(20# o 2 ) = y(20 x 19-73") = v(394-6) = + 19-86". The value of 

 the whole error is consequently + v(408*7 + 394-6) = + 28-34". 

 This is to be divided by 100, thus the result by II. may be ex- 

 pressed 89° 59' 52-7" + 0-3". 



It is seen by these determinations that the value of repetitional 

 is very much greater than that of independent measures. The 

 probable error of the mean of nineteen independent measures such 

 as shewn in I. would be 1*40", as contrasted with 0*38" the actual 

 probable error. Further that the probable error of the method in 

 II. is greater than if the ten series were independent, the errors 

 being in the example 0'28" as contrasted with 0-21". 



Tabulated probable errors, and empirical combinations of measures 

 made with different instruments. 



The reading of the graduated circle after each measure, although 

 affording the means, generally, of obtaining a better value for the 

 angle, and a check upon any erroneous use of the instrument (as 

 say the movement of the wrong tangent screws) involves so much 

 time, that it has become a common practice to note the first 

 measure, and the final reading only, dividing the latter by the 

 number of measures. The result should approximately agree with 

 the recorded first measure, which serves therefore as a safeguard 

 against error in respect of that number. It has been usual to 

 assume that the probable error of the result is reciprocally pro- 

 portional to the number of measures, to which ratio indeed it 

 closely approximates when the probable graduation (or reading) 

 error is very large as compared with that of pointing. How far 

 this assumption is justifiable is exhibited in the following table 



