486 'Journal of Comparative Neurology and Psychology. 



find from the correlation table the diameter of the nucleus corre- 

 sponding to the any given diameter of the cell-body. The value 

 of the nucleus thus obtained is however affected by a variable 

 probable error owing to insufficient number of observations com- 

 bined with a random sampling. We therefore need to find the 

 most probable values from the observed data, or the characteristic 

 equation which can best represent the data with minimum error. 

 We have two kinds of characteristic equations, linear and non- 

 linear. Whether or not a given expression can be best represented 

 by the linear or non-linear characteristic equation is of the utmost 

 importance, and it is necessary to determine which equation 

 applies to our present data. Pearson ('04) has introduced a 

 new constant, tj, called the correlation ratio and this is used to 

 test the linearity of the regression. The correlation ratio accord- 

 ing to Pearson is the ratio of the variability of the means of the 

 arrays of one correlated character to the total variability of that 

 character and is shown in the following formula: 



The constant tj has the same value as the coefficient of correla- 

 tion when the regression is perfectly linear. If the regres^on is 

 not linear i) will be greater than r. Then evidently rj^ — -r is a 

 measure of the approach of the regression to linearity. I have 

 calculated the value of i) by the formula given above and found 

 that when this value is compared with the coefficient of correlation 

 the former is significantly greater than the latter as is shown in 

 the following: 



Tj - r = .9267-. 8616= .0651 



However, the difference between the value of these two constants 

 will in practice deviate more or less from zero. It is therefore 

 necessary to find whether or not the difference found between the 

 two constants is significant. Recently Blakeman ('05) has given 

 methods of obtaining the probable error of various functions 

 ofrj^ — .r. If we let 



C = f-r^' 



an approximate formula for the probable error of ^, i.e., E , is 

 £„ 0.6745' ^ ^^ i/i + (i_^2)2_(i_r2)2 



