180 Dr. Norman Campbell on the 



the true values. The rule for applying this method to the 

 third problem may be stated as follows : — 



It is assumed that the equation which the measured 

 magnitudes have to satisfy is linear and of the form 



aoc + ly + cz+ . . .+m = 0. .... (1) 



If (as in the example of radioactive decay) the equation is 

 not of this form, certain methods (which will be accepted 

 for the present without inquiry) are available for reducing 

 it to this form. It is clear also that, without loss of 

 generality, we may always put a=l, and this procedure 

 will be adopted in what follows. If there are n variables 

 (#, y, z, . . .) and, consequently, n constants (b, c, . . . m),. 

 and if N sets of the variables have been measured, we have 

 N equations of the form 



X\ + by l + cz 1 + . . . 4- m = 0," 



x-s-h bt/js+ (:^-f . . . ' +m == 0. 



(2) 



N is greater than n. To obtain unique values of (b, c, . . . ?>/) 

 we have to reduce these N equations to n equations. We 

 form one of these equations by multiplying the rth equation 

 by x r and forming the sum of all the equations so multiplied ; 

 another by multiplying the ?*th equation by y r and forming 

 the sum ; another by multiplying the rth equation by z r ; 

 and so on. We thus obtain n "normal" equations relating 

 the n constants (b, c, . . . m) to sums of squares and products 

 of the variables (.i\ y,-z f . . .). In these equations (b, c, . . . in) 

 are now treated as variables; the solution of them gives the 

 true values of (6, c, . . . m) . 



The rule for solving the second problem can be expressed 

 in a form very similar ; but since, as has been noted already, 

 this problem has not much physical importance, it will be 

 left on one side for the present. It will concern us only in 

 so far as we have to determine whether any other principles 

 proposed for solving the third problem are, like those of the 

 Method of Least Squares, also applicable to the second. 



Regarded apart from the theory of errors on which it 

 professes to be founded, the rule given is merely a device 

 for reducing the K equations for n unknowns, the solution of 

 which must be indeterminate, to n equations for n unknowns, 

 the solution of which is determinate. But there is a much 

 simpler method of effecting the reduction. We may simply 

 divide the N equations into n groups, and add all the equations 

 in each group : we thus arrive at n " normal " equations. 



