412 MR. K. PEARSON ON THE MATHEMATICAL THEORY OF EVOLUTION. 
Nove. 
Added May 24, 1895. 
[Since writing the above memoir my attention has been drawn to a note in 
Dr. WesterRGAARD’s “Theorie der Statistik,” referring to Professor T. N. THrete’s 
treatment of skew frequency curves. I have procured and read his book, ‘ Forelaes- 
ninger over Almindelig lagttagelseslaere, Kj¢benhavn, 1889. It seems to me a. very 
valuable work, and is, I think, suggestive of several lines for new advance. It does 
not cover any of the essential parts of the present memoir. Dr. THIELE does indeed 
suggest the formation of certain “ half-invariants,” which are functions of the higher- 
moments of the observation—quantities corresponding to the py— 3p", bs — 10p.p5, 
&ec., of the above memoir. He further states (pp. 21-2) that a study of these half- 
invariants for any series of observations would provide us with information as to the 
nature of the frequency distribution. They are not used, however, to discriminate 
between various types of generalised curves, nor to calculate the constants of such 
types. A method is given of expressing any frequency distribution by a series of 
differences of inverse factorials with arbitrary constants. Thus if 
and 
AB,, (x) = B,, (x + 4) —_ [st (x te 3) 
we can express any law of frequency y = f(x) by 
SF (&) = boBn (#2) Fb, AB (@) +.» + bn A"B) (2), 
where the constants bo, b,... 0, can be determined numerically when the frequency 
of n + 1 chosen derivation-elements is known. 
I see a possibility of more than one theoretical development of interest, especially 
in relation to compound material, from this development of Dr. TurELE’s, but I doubt 
whether it can be of practical statistical service even as an empirical expression for 
frequency. Instead of having the 8 to 5 constants of our generalised curves, the full 
value of Dr. THrexe’s expression requires as many constants as there are recorded 
frequencies, and then expresses the result in functions like A’B, (x), by no means easily 
realised or likely to appeal to the practical statistician. It is true the complete series 
gives absolutely accurately the frequency of all the points used in the calculation, but 
it does not, like the generalised curves, indicate the purely accidental variations of 
the frequency. If, on the other hand, we take, as Dr. THIELE suggests, some half- 
dozen terms only of the series—which give the really essential character of the 
