POSITION REGRESSION— PRIMARY BRANCHES, 65 



We have to determine the form of the function ^. It is obvious that 

 the proper way to set about this is to find out what curve best graduates 

 the observed data, and the equation to this curve will be the expression 

 of the law of growth v/hich we are seeking. While the problem is thus 

 theoretically a simple one, practically it is an extremely difficult one, 

 because if the data give no clue at the start as to what the nature of 

 <^ (x) is, which unfortunately is usually the case, we have to resort to a 

 very laborious process of trial and error. Different curves must be 

 fitted one after another to the data until the right one is found. It is 

 the usual custom among physicists to assume a parabola and get the 

 best fit possible by increasing the number of constants. This method 

 is not, however, a theoretically justifiable one, because, as Pearson 

 (:02, p. 19) has pointed out: 



There are often considerations, lying outside the actual data, which suffice to indi- 

 cate that trigonometrical, exponential, or other types of curves will give better results 

 than parabolas. A parabola which passes even through all the observations may 

 indeed be a most undesirable representation of the facts, for it has twisted and curled 

 to account for error as well as to give the general sweep of the observations. 



The figures in table 34 and the diagrams in fig. 11 show clearly that the 

 nature of the regression curve is the same for all the series, and that 

 further, the absolute values for Series I, II, and III are, within the limits 

 of error from random sampling, identical. Hence on account of the 

 very laborious character of the arithmetic involved, these three series 

 (I, II, and III) were combined, and in what follows this combined mate- 

 rial will serve as a basis. By this proceeding a considerable gain in 

 smoothness of the figures to work from was made, without any loss 

 of accuracy. The means for the combined series are given in the fifth 

 column of table 34. 



Having at the outstart no idea of the nature of </> (x) , it was decided 

 as a beginning to fit a series of parabolas to these data. That is to say, 

 in the expression 



it was assumed that 



<li{x) = c„ +A^ + c^x"" + Cgx' + . . . cX- 



By taking more and more terms of this expansion we get successively 

 higher order parabolas. There is no theoretical limit to the process, 

 but obviously, if we are seeking a true graduation formula, it is idle 

 to go higher than the sixth-order term, for the reason mentioned 

 above. Furthermore, to get the sixth-order parabola, we require 

 moments up to and including the sixth, and as Pearson (:05 and else- 

 where) has shown, the probable errors of the higher moments rapidly 



