176 jNIr Sang on an erroneous Method 



struments would be zero. In order to obtain a measure of the 

 inaccurac)^, we must average the squares of the errors: this 

 average never can be zero unless the divisions be absolutely 

 perfect. The sum of the squares of the errors also depends on 

 the arbitrary position of the normal system ; and it is clearly 

 fair so to choose this, as that the measure of inaccuracy be the 

 least possible. This occurs when the sum of the errors is zero. 

 Assume, then, the position of the system of exact divisions, 

 so that 2/? = o : /?'^ 4- 2''^ + r^ 4- &c. or 2 .p* will be the mea- 

 sure of entire inaccuracy, and - 2 ;?^ that of the mean inaccuracy 



to be expected from a single observation ; representing this 

 expectation by {e^Y^ we have 



The value of this expression depends on the peculiarity of 

 division, and can only be ascertained for each instrument by 

 direct experiment. 



We have now to inquire what is the probable error from an 

 observation with two readers. The errors of the two readings 



being p and g, that of their mean must be ^-^-^; therefore, 

 supposing the errors to be scattered by chance round the limb, 

 4- 2 . (/? + 9)^ is the entire measure of inaccuracy arising from 

 double reading. This may be put under the form 



in which p^ will occur 2 {n — 1) times, ^ pq twice ; hence the 



total inaccuracy is ~ 2.p^-f2p^. Here I observe, that 



-|^2.p^-|-2/?^ = ^ {2pY = o ; so that the measure of entire 



inaccuracy is -^—2/?^. Now, the entire number of chances is 



n{7i — 1), hence the measure of inaccuracy on the double 

 reading is 



^/ n n — 12 ^ 

 n — \ 2 v^'^ 



