with Rtmarks on the Law of frequency of Error. 45 



specimens, ought to be looked upon as a multitude of different 

 sorts mixed together. The proportions inter se of the different 

 sorts may be accepted as constant ; there is no difficulty arising 

 from that cause. The question is, why a mixture of series ra- 

 dically different, should in numerous cases give results apparently 

 identical with those of a simple series. 



For simplicity's sake, let us begin with considering only one large 

 influence, such as aspect on the size of fruit. Its extreme effect 

 on their growth is shown by the difference in what is grown on 

 the north and south sides of a garden-wall, which in such kinds 

 of fruit as are produced by orchard-trees, is hardly deserving of 

 being divided into more than three phases, "large, " "mode- 

 rate," and " small." Now if it so happens that the " mode- 

 rate" phase occurs approximately twice as often as either of 

 the extreme phases (which is an exceedingly reasonable suppo- 

 sition, taking into account the combined effects of azimuth, 

 altitude, and the minor influences relating to shade from leaves 

 &c), then the effect of aspect will work in with the rest, just 

 like a binomial of two elements. Generally the coefficients of 

 {a + b) n are the same as those of (a + b) n ~ r x (a + b) r . Now the 

 latter factor may be replaced by any variable function the 

 frequency and number of whose successive phases, into which 

 it is necessary to divide it, happen to correspond with the value 

 of the coefficients of that factor. 



It will be understood from what went before, that we are in a 

 position to bring these phases to a common measure with the 

 rest, by the process of fictitious grouping with appropriate doses 

 of minute influences, as already described. 



On considering the influences on which such vital phenomena 

 depend as are liable to be treated together statistically, we shall 

 find that their mean values very commonly occur with greater 

 frequency than their extreme ones ; and it is to this cause that 

 I ascribe the fact of large influences frequently working in toge- 

 ther with a number of small ones without betraying their pre- 

 sence by any sensible disturbance of the series. 



The last difficulty I shall consider, arises from the fact that 

 the individuals which compose a statistical group are rarely affected 

 by exactly the same number of variable influences. For this cause 

 they ought to have been sorted into separate series. But when, 

 as is usually the case, the various intruding series are weak in 

 numbers, and when the number of variable influences on which 

 they depend does not differ much from that of the main series, 

 their effect is almost insensible. I have tried how the figures 

 would run in many supposititious cases ; here is one taken at 

 haphazard, in which I compare an ordinary series due to 10 

 alternatives, giving 2 ,0 =1024; events, with a compound series. 



