Adjustment of Observations. 227 



Eliminating y^ y 2i y 3 . . . we obtain easily 



(5»-j>)2*y = 6(2*' a -pSj' 3 ), 



or putting ^ t x' 2 —pX//' 2 = 2m%x'y', 



b 2 -2mb-p = 0; 

 whence 



6 = m + v ?ji 2 +j>, 



the negative value being obviously inadmissible. 



Had the relation between x and y been written ax—y, 

 a being the specific volume, we should have obtained by an 



exactly similar process a=-(— m + s/m 2 -rp), which is equal 



1 ^ 



to j. Thus the two methods of attack give identical results, 



as should be the case. 



It follows from the form of the result (as is indeed evident 

 a priori) that to obtain an intelligent solution we must know 

 the value of p, that is, the relative accuracies of the measure- 

 ments of mass and volume. If p is infinite (measurements 



of volume exact) b= ^ ,„ , while if p is zero, or rather 



1 . . . 2 ' y tx'* 



- infinite (measurements of mass exact) b = ^ — j-,. These 



p %.vy 



are the two results given by Dr. Campbell, and are of course 

 in general, as is to be expected, different; when p has a 

 finite value, that is, when both sets of measurements are 

 fallible, the value of b lies between these two extremes. 



$ 



nnmary. 



Dr. Campbell's article serves a useful purpose in drawing- 

 attention to the fact that a rigorous least squares solution is 

 sometimes laborious, and that more approximate methods 

 will often yield results sufficiently accurate for most pur- 

 poses. On the other hand, many of his statements in regard 

 to the principles of the method of least squares are erroneous, 

 and his application of them to several classes of problems 

 which he considers is incorrect. The method of solution 

 for the general case, including as particular cases all the 

 problems treated by him, is indicated, and is applied to two 

 specific examples. 



Dominion Observatory, Ottawa, 

 March 4th, 1920. 



