METHODS 189 



but with practice, it is not difficult. The final result is that the co- 

 efficients of the equation are obtained, and the result of months of 

 work can be written in one line. 



Joint Functional Regression Diagrams 



These require the variables to be sorted in three ways, first according 

 to the two independent variables, as just described, and then the mean 

 spawn date for each bundle has to be found. The analytical processes 

 are illustrated in Figs. 47. The data are taken from the joint functional 

 regression diagram for Mi (Fig. 36). In order to reduce the space 

 needed for a full reproduction of the processes involved in constructing 

 the diagram, only the data from 32°F to 38°F and from o to 70 mm of 

 rainfall are used. This has required only 22 curves, instead of the 72 

 much larger ones in the actual investigation, but they are sufficient for 

 the explanation. The removal of part of the data has naturally altered 

 some of the details, and Fig. 47 d is therefore not quite the same as the 

 corresponding part of Fig. 36. In Fig. 47 a the original data arc 

 indicated by dots, each one of which is the mean of all the records with 

 the corresponding co-ordinates. It happens that these means were all 

 derived from more than ten records, but, if any mean had been based 

 on less than ten, the actual number would have been written alongside 

 the point as a guide to the weight to be attached to it when drawing 

 the line. Through these dots, a line has been drawn free-hand, lying 

 as evenly as possible among them, as far as can be judged by eye. In 

 47 b is shown a series of graphs at lo-mm intervals of rainfall, in 

 which each point shown has been read /rom the curves in {a). For 

 example, the point on the 45-mm curve at 34°F in 47 b is exactly at 

 70 days, and it can be seen from the curve of 34°F in. 47 a that the 

 curve cuts the 70-day line at 45 mm. In 47 c a similar operation has 

 been performed using the curves of 47 b as data, plotting against 

 rainfall as abscissae. These curves therefore reproduce those of 47 a, 

 but have been smoothed and adjusted by taking account not only of 

 the points along each line, but according to the neighbouring lines as 

 well. As can be seen, the irregularities of 47 a have been removed, and 

 the whole series appears as a regular family of curves, changing in form 

 gradually from curve to curve. Fig. 47 J is a synthesis of all the curves 

 , in 47 c, produced by selecting the dates 60-80 in 5-day intervals, and 

 finding the temperature and rainfall that correspond. These co- 

 ordinates are plotted and the points joined to form the isophenes. 



