THEORY OF ERRORS. XCV 



of ^x, Ax,, . . . from (<r) in (a), whereby there will result I: equations containing 

 the Qi, Qo . . . Qic alone as unknowns. The result of this substitution is 



[f] +m +[f] + ■+-=°- 



These equations {d) are derived directly from {c) in the following manner : multi- 

 ply the first of {c) by ^'' the second by -"' etc., sum the products, and compare the 



sum with the first of {a). The first of {d) is then evident j the others are obtained 

 in a similar way. 



The mean error of an observed quantity of weight unity is in this case given by 

 the formula 



'"' - V k" ' 



where k is the number of conditions (a) ; and the mean error of any observed 

 value of weight/ is 



sjp- 



f. Computation of mean and probable errors of functions of observed 



quantities. 



Let V denote any function of one or more independently observed quantities 



X, y, z, . . .\ that is, let 



V=f{x,y,z...). 



A question of frequent occurrence with respect to such functions is. What is the 

 mean * error of V in terms of the mean errors of .r, y, z, . . . "i The answer to 

 this question given by the method of least squares assumes that the actual errors 

 (whatever they may be) of x, y, z, . . . are so small that the actual error of Fis a 

 linear function of the errors of x, y, z. In other words, if e„ e^, e., . . . denote 

 the actual errors of .r, y, z, . . . , and A Fdenote the corresponding actual error of 

 V, the metliod assumes that 



wherein the squares, products, etc., of e„ e^, e„ . . . are omitted. 



This condition being fulfilled, let c denote the mean error of V, and e^., c,,, c^ . . . 

 denote those of x, y, z, . . . respectively. Then the law of error of least squares 

 requires that 



* Since the probable error is 0.6745 times the mean error the latter only need be considered. 



