444 
PEOFESSOR K. PEARSON ON MATHEMATICAL 
biological investigations. It seemed, therefore, desirable to enter a little more 
fully into the analysis of the cases in which the criterion was positive but v 
imaginary, and discover what types of frecpiency curves had escaped my attention.^ 
The key to the solution lies in the fact noted on p. 369 of the memom, namely, that 
even if the criterion be positive, there will still be a solution akin to Type I. and not 
to Type IV. if e be negative. No frequency series satisfying these conditions had at 
that time come under my notice, and later, when collecting data of floral variabilitv*, 
my own remark as to e had slipped from my memory. It is the object of this 
supplement to obtain an improved criterion of type, to discuss the nature of the 
curves which fill the gap observed, and to illustrate by one or two examj^les the 
fitting of such curves to actual statistics. 
(2.) The Two Criteria. 
Throughout this supplement the notation of the previous memoir will be assumed 
to be familiar to the reader. 
Turning to p. 378 of that memoir, we note that since and r — 1 are necessarily 
joositive, s if positive must be > rb Hence v can only become imaginary if s be 
negative, or 
/g, - 2)- 
16 (r- 1) 
> 1 . 
Substitute in tbis tbe value of r and it becomes 
A (A + fd' _ 
3^h(-/g2 — 3^1 — <J) 
(ii.). 
Hence the complete condition that a curve of Type IV. shall give the distribution 
of frequency is not only 
= 2/3. - 3/3i - 6 > 0, 
but also 
_ _ /S, (/3. 4 _ 
“ 4(4yS2 - 3/9,) (2^, - 3/9i - 6) ^ 
Turning back to p. 369, we see that e heing positive the complete conditions for a 
curve of Type I. giving the distribution of frequency are 
Ky = 2/3. — — 6 < 0, 
* I was very loath to adopt Professor Edgeworth’s method of inventing new frequency curves by 
putting X = f{x') in a normal frequency distribution, y = Besides strong theoretical objections to 
this process, I had found Equation (i.) so sufficient for a great variety of cases that I felt confident it must 
cover the newfi- discovered outstanding cases, and this confidence seems justified bj^ the result. 
