MR. K. PEARSON ON THE MATHEMATICAL THEORY OF EVOLUTION. 361 





pn (pn — 1) (pn — 2)... (pn —7 + 1) 
n(n — 1) (n— 2)...(n—7 + 1) 
é qu r(r — 1) gn (qu — 1) 
eet 1.2 (pn—r +1) (pn—r + 2) 
7 (7 — 1) (7 — 2) gn (qn — 1) (qn — 2) ): 
u 1A) (pu — 7 + 1) (pr —7 + 2) (Gaapeay Te 
If y, be the s ordinate of this polygon, and we suppose these ordinates plotted up 
at distances ¢ apart, we have 
pen C= 8 ap EG 8 sp i 



Fan aie es 
LSC. Cera stew li) ces 
Xu, 0 (s+ 4). 
Thus 
Ys+1 — Ys 2 (y +1) (1 + qn) — 8 (n + 2) 
E Wert yy xe 6 (+1) +n) —s{20¢ 41) +2(q—p)} + 28 
7+1)a+q)—- (A — t) (n + 2) 


\ Q 
“@ +d tom — (M93) BO 41 + m(g— pr 2( = §) 
\ ¢ G é 
Write 
E (+10 + a) 
Xiey = Key 0 (9+ n+ 2 } 

and we find with our previous notation 
Ay 1 aa Xi s43 ( ) 
= = =a ay eatclste om Lata oceaeeesceeeeine 1 ((E))) 
Av Ys43 By + BiX’s4y + BsXs45 

where 
a 2 (7 +1) (w~@—7r4+1) (1 + qn) (1 + pn) 
Ua (n + 2)3 ‘ 



__ on (n — 2r) (p—4@) ee ELS 
= gaan |. * Pa ae 
Now, if we attempt to find the curve which has the same geometrical relation for 
the slope as the above hypergeometrical polygon, we see that it will change its type 
according to the sign of 8,” — 48,63. 
After some reductions we have 

Vesa ieee eee og Lat 
pe Ver ee a tegae V et seta) 
MDCCCXCV.—A. ; 3A 
