﻿III. 



On the Use of Equivalent Factors in the Method of Least Squares. 



By GEORGE P. BOND, A.M., 



ASSISTANT AT THE OBSERVATORY OF HARVARD COLLEGE. 



{Communicated April 15, 1856.) 



One of the most important applications which has been made of mathematics to 

 investigations in physical science has for its object to ascertain the best manner of 

 combining data affected by unknown errors of observation, so that the probable effect 

 of these errors shall be the least possible. The method of least squares proposes 

 to accomplish this, by reducing to a minimum value the sum of the squares of the out- 

 standing errors, and, by conforming to this single criterion, to fulfil the condition, so 

 desirable in the prosecution of thorough and exact research, of reducing to its least 

 possible amount the influence of errors in the data employed. 



The investigations here presented have been entered upon ^vith the design of deter- 

 mining the degree of numerical exactness proper to be observed in making use of the 

 method of least squares, in order to secure its peculiar advantages with the least out- 

 lay of labor. 



Some detail in the discussion seems to be called for from the prevalence of a practice, 

 almost universal among computers, of adhering to the letter of the method of least 

 squares with a strictness which implies a misapprehension of its true spirit. It is 

 impossible to adduce any valid reasons to justify such a course when it must be fol- 

 lowed at a serious expense of time and labor ui the computations. 



It has not escaped the observation of Gauss, m his original exposition of the method, 

 that some freedom of interpretation may be allowed when its theoretical results are 

 applied in practice, as the following passage, referring to the solution of equations by 

 least squares, will show : — 



