CONTRIBUTIONS TO THE THEORY OF EVOLUTION. 
445 
and 
4 ( 4 ^, - 3/3,) ( 2/32 -S/3,-6)^ ' 
The latter condition will be always satisfied since /3, and 4^o — 3^, are positive for 
any distribution wdiatever, and 2f3o, — o/3, — 6 is negative by hypothesis. 
Further, in tlie previous case is seen to be essentially positive. 
Hence the criteria written down cover all possible cases but those for which 
Ko > 1. 
Sub-cases which arise from transition curves just at the limits will, however, be 
likely to be of interest, What happens when k-j = oo and when Ko = 1 ? The only 
possibility for <x> is 2^.^ — 3/5, — 6, or k, = 0. But this curve has been fully 
treated under Type III. in the memoir. 
We shall see later that Ko = 1 leads us up to a novel transition curve of consider¬ 
able interest. 
To ascertain something about the general case in which > 1, let us return to 
the memoir again and examine the value of e on p. 3G9. It can only be negative if 
I + i/^i (^’ + + 1) < 0) 
where r is here 
Substituting, we find at once 
6 (^ 2 -/ 3 , - 1 ) 
3 / 3 ,- 2 ^.^ + 6 ' 
K,> I, 
which in itself involves k, > 0 . 
Hence the missing gap corresponds to those cases in which e is negative. 
It will be clear that although in form giving a more complex criterion than k,, 
is really more effective, as covering all the possible cases. We have then the 
following scheme : — 
Criterion k.,. 
Corresponding frequency curve. 
K.) = ^ . 
Transition curve, Tj'^pe HI. (Memoir, p. 373). 
K'2 > 1 & < CO 
Type VI. (see p. 448 below). 
K., = 1. 
Tran.sition curve. Type V. (see p. 446 below). 
K2 > 0 & < 1. 
Type IV. (Memoir, p. 376). 
k.2 = 0, /S, = 0, /3.2 = 3. 
Normal curve. 
Ko = 0, /3, = 0, /3-2 not = 3 
Type H. (Memoir, p. 372). 
K., < 0. 
Type I. (Memoir, p 367). 
