THE CORRELATIONS OF AREAS OF MATURED CROP AND THE RAINFALL. 353 



In this paper the crops considered are those which are known technically as 

 ' unirrigated,' l that is, they do not ordinarily receive any water except that which falls 

 on them directly in the form of rain. No doubt crops which receive water from other 

 sources, such as canals, wells, or by spill, 2 from torrent or river, depend, too, very largely 

 on the water which reaches them directly in the form of rain, and in any case they 

 will in general depend ultimately on the rainfall which falls within a certain distance; 

 it might therefore be reasonably anticipated that the amount of such crops would 

 show a correlation with the local rainfall, but the correlation will probably be smaller 

 than that for the class of crops dealt with here, and its determination will be left to 

 a future date. 



It would be higrny desirable that every different kind of crop should be treated 

 separately. In the present instance the large probable errors to which most of the 

 constants are subject make it open to question whether the differences in the coeffi- 

 cients of correlation and regression would be large enough to be significant. Though, 

 undoubtedly, this is of importance, it is not proposed to consider here the constant 

 special to each class of crop. This point must be noted when applying the regression 

 equations for purposes of prediction to any particular group of unirrigated crops, since, 

 should the group taken consist of different classes of crops from those whose regression 

 equations are given, or consist even of the same staple crops in different proportions, 

 it is obvious that the results obtained would be greatly diminished in accuracy should 

 the differences in the constants mentioned above be really significant. 



Again, caution must be used in attempting to apply the equations based on the 

 special circumstances of one locality (not only in respect of the nature of its crops) to 

 a locality of completely distinct physical and climatic conditions. Thus, to take 

 an important case, it would clearly be unjustifiable to assume that the equations would 

 remain unmodified if they were determined for a locality in which the distribution of 

 rainfall differed greatly from that which is characteristic of the localities discussed, 

 even although the total mean rainfalls might be the same. 



§ 2. The general process adopted in dealing with the statistics of rainfall and 

 crop is as follows : 



Let C denote the known matured area of crop in any village, group of villages, or 

 other arbitrarily chosen area for a given harvest. Let R denote the total rainfall 

 during a given period measured at any selected point, preferably as near as may 



values of the x's and y's respectively, so that if N be the total number of associations Nx = ^(x r ), Ny = 2(y r ) where s 

 denotes a summation for every value of x and y respectively, then the 'standard deviation' of the x - variable 

 which is usually denoted by <r x is given by NcrJ i = 2(xn-x) 2 . A similar expression holds for the y— variable. Then if we 



write NI=2(x n -z)(y n -y), where the summation is for every associated pair of x's and y's, the correlation coefficient 



I 



is given by r = . 



<r a 



x y 



The standard deviation is readily seen to be a quantity of the same quantity as the ' radius of gyration ' of dynamics 

 or the R.M.S. of the electrician. The quantity / corresponds to the dynamical ' product moment ' divided by the 

 mass. 



l A poor equivalent for the Urdu word ' barani ' or rainland crops. 



i These are also frequently classed as ' unirrigated,' but they have been excluded from this investigation, as they 

 are quite differently circumstanced. 



