Correlation of Averages" for Four Variables. 



373 



are loci of equal frequency, analogous to Mr. Galton's ellipses 

 and Professor Edgeworth's ellipsoids; the former of which can 

 easily be observed as loci of equal height on a probability- 

 surface, the vertical being taken as the dimension of frequency. 



Professor "Weldon's measurements of the organs of shrimps 

 supply a real biological example to which can be applied for 

 more than three variables the principles of correlated averages. 

 The theoretic, no less than the practical, interest of this 

 example is much enhanced by the fact that specimens from 

 the Plymouth and from the Naples coast were measured, the 

 number of measurements being 1000 in each case. Thus 

 excellent material exists for the computation of the quantic 

 pertaining to two distinct groups of the same species, and the 

 two sets of coefficients thus calculated may be compared. 



Selecting from Professor Weldon's ^-coefficients the 



six. 



correlating four organs, that appear to be most suitable, we 

 have for our data the following systems : — 





Pvr Pis- 



Pur 



P23- 



P-2i- 



P 3 f 



Naples 



•29 



-•23 



•27 



•71 



•76 



•G3 



Plymouth ... 



•24 



-•18 



•22 



•78 



•78 



•67 



It is proposed in the following pages to calculate the two 

 quantics to which these systems of coefficients respectively 

 give rise, as examples of correlation between the averages of 

 four variables. 



Since p 12 and p 21 have the same meaning, and a similar 

 remark applies to all the coefficients, it is allowable to change 

 the order of the numerals for the sake of symmetry whenever 

 it is found convenient to do so, there being in each case an 

 order which is best for securing the immediate detection 

 of a mistake. Also it is often of assistance to write p n , p 22 , 

 &c. for unity, as this makes the order of the numerals more 

 apparent. 



Let A, as usual, be the discriminant of R, i. e. 



A='| pi £i 2 qu q H 



<M P-2 ?23 724 

 </31 ?32 PS 734 

 741 742 743 Pi 



* Them's here might, for symmetry, be called q lv q 22f q 33) q u . 



