302 PROFESSOR K. PEARSON AND MR, L. N. G. FILON 



Or, taking logarithmic; differentials, 



AS JS, = cot < A< - (^ - i^) Ar, 

 whence, 



SI/SI = C0t s 0$ + ' ' 



In a similar manner Ej Sl and R aSli can be found if desired. None of these quantities 

 will, as a rule, vanish, and as very many measurements on animals give curves of 

 the tangent type, we conclude that in general all selection of the size of an organ 

 alters its variability and the sketvness of its distribution, and again all selection of 

 variability connotes alteration of the size and skewncss of the selected organ. 



The probable errors of p. 2 , p. A , and p. t , as well as the error-correlations of these 

 quantities, can all be found from the differentials of (cxxiii.), (cxxiv.), and (cxxv.) ; 

 the calculation is laborious, but presents no novelty. 



Lastly, the probable errors of the mean and modal frequencies may be deduced. 



For the mean frequency we start from (cxxxv.) and use (cxxxiv.). 



This requires us to know AX, where 



' ' C S 2S 



We have, as in (cxlvii.) and (cxlviii.), 



cos 



s ^ sin 2 ^ 



where c, and c-, admit of fairly easy calculation. 

 Hence, by (cxxxv.), we find, 



+ (i + siir <^> + c,) Ar/r (c 2 + cos <f> sin <) 



a* + (i + si" 2 <^> + o,) 2 S;/7- 2 + (c 2 + cos <j> sin ^) 2 



2 / 1 2 j \V-C-D . 2 (c 2 + cos ^> sin </>) 



-- (| + sm 2 <j> + c,) 2 a S,.Ra, + ~ - S 



i fL tt/ 



sin a + <g) (c 2 + cos < sin <>) 



If the problem be to find the modal frequency y-, 8x, we easily deduce y-, by putting 



