184 



Dr. Norman Campbell on the 



where (X, Y, Z, . . .) are the means * of a group of observed 

 The solution is 



b = D»/D ; C = D c /D ; . . . ; m = D w /D ; . (4) 



r here 



D* = 



XxZj. 



. . 1 



; D.= 



Y x Zi • • 



.*1 



: D = 



YxZx. 



. . 1 



■J\:2 ^2 • 



. . 1 





Y 2 Z 2 . 



x 2 





Y 2 Z 2 . 



. 1 



• • 



• 





• • 



• 





• • 



• 



Now suppose that only one of the measured magnitudes, #, 

 is liable to error amd that all the measurements on the others 

 are absolutely correct. This assumption is practically true in 

 a large number of important cases ; it is moreover essentially 

 involved in the method of L.S., so that we are not introducing 

 any new error by adopting it. Then if db, de, . . ". dm are the 

 errors caused in the calculated values of "6, c, . . . m, by errors 

 dX l5 <iX 2 , ... in X 1? X 2 , . . . 



(5) 



db = l/D(rfX x . W + dX 2 . IV+...), ^ 

 dm=l/D{dX l B m 1 + dX 2 . 7 D 7] ; 2 +...), J 



where D b r is the minor of X r in D^,. 



The mean error of L.S. (from which the probable error 

 is calculated by multiplying by 0'6745j is the mean of all 

 the errors which might be expected if the observations 

 were repeated a large number of times and the quantities 

 calculated from each set of observations. In deducing it, 

 it is assumed that the mean square errors dXi 2 , dX 2 2 , . . . 



are all equal and that the mean product errors dX b dJL 2 . . • 

 are all zero. Adopting this assumption, we find 



db 2 = dK 2 /p b ; ~dV 2 =~dX 2 / Pc ; . . . ; din 2 ^~dX 2 /p m ; (4) 

 where 



i/p, = 



(wy+(D, 



\r- 



(5) 



* Strictly speaking, they are the sums, not the means. If q is 

 they could be converted into means by dividing each of the equations 

 by the same number p — without, of course, any effect on the result. 

 If q is not the results would be slightly different if some equations 

 were divided by p and some by p + 1, but again the effect of neglecting 

 this difference is quite inappreciable. In practice the normal equations 

 will always be sums, not means ; but in discussing them generally it will 

 be assumed that they are means, not sums : no appreciable error is 

 thereby introduced. 



