304 Prof. K. Pearson. Mathematical Contributions [Jan. 20, 



may use the general theory of partial correlation, otherwise we must 

 fall back on results like (x). For example, in head growth in boys, 

 we cannot much improve on a straight line ; in positional influence on 

 the branches in the whorls of Equisetum arvense we need at least a third 

 order parabola. 



(8.) Although material for several investigations on the homo- 

 typosis of serial homologous parts has been collected, the progress in 

 some of these cases is slow, as it involves rather laborious micro- 

 scopic measurement. I content myself at present with an illustration 

 from the vegetable kingdom. 



I collected in the autumn of last year, 126 plants of Equisetum arvense 

 in Eaydale Side, an offshoot of Wensley Dale ; the plant was growing 

 on a lane side high up above Semmer water. This Equisetum. grows from 

 the top with a single stem, and I counted the number of branches to 

 the whorl from the root upwards. As a rule, there will be one or 

 two whorls close to the soil which have never developed any branches 

 at all ; then we have what I shall term the first whorl in which some 

 branches have developed, but the number is irregular and obviously 

 subject to some cause of variation, other than the growth law of the 

 plant. The number of branches to the whorl then increases uniformly 

 and steadily up to the 4th whorl, after which it falls almost equally 

 steadily to the 10th whorl. Beyond this the results becomes somewhat 

 irregular again, for very few plants will be found — -at any rate in the 

 locality considered — with more than 12 or 13 whorls, and even in these 

 whorls there is a certain amount of forking or irregularity which it 

 is difficult to deal with. The plants were certainly fully developed 



be linear, the means of the arrays all lie on the regression line, and the mean 

 standard deviation of the arrays about their means is trVM — r 2 . If the regression 

 be not linear, the means of the arrays will have a mean square deviation 2m 2 from 

 the regression line. The mean square deviation of the arrays from the regression 

 line, but not from their means, is still a 2 (I — r 2 ). The mean standard deviation 

 (deviation of mean square from means) is now given by 



since a' 2 = cr'' 2 + vyf- But we easily find 



Hence 2m is a good measurement of the deviation of regression from linearity, or 

 of <r M from ro. If we take r] = <tm / <r, we have 



«r'2 = 0*0— * a )i 2 M 2 = {n 2 -r 2 )<r\ 



Clearly rr must lie between r 2 and 1. Further, i\ can only vanish when the cor- 

 relation is zero, or become ±1 when the correlation is perfect. Between these 

 values it gives the mean reduction in variability of an array as compared with the 

 whole population. Further, the deviation of q from r is a good rneasur-e of the devia- 

 tion of the system from linearity. Thus rj is a useful constant which ought always 

 to be given for non-linear systems. It measures the approach of the system not 

 only to lineaiity but to a single valued relationship, i.e., to a causal nexus. 



