352 MR. K. PEARSON ON THE MATHEMATICAL THEORY OF EVOLUTION. 
a = 171°6, fig = 10°14, 
fy = 15°95, ju, = 826°34, 
all in centimetre units. 
These give 
B, = °24401, By = 3°1739. 
Hence for trapezia, 
*38422° — °7499172? + 0180082 + 003389 = 0, 
and for rectangles, 
*384223 — °874962* — -003832z + 000424 = 0. 
These give the following solutions :— 







Trapezia. Rectangles. Lines. 
Z 192516 2°28034 2°6028 
a 19°379 23-983 28°5293 
p 8881 "8936 89985 
q “1LLO 1064 ‘10015 
C 2°2017 20712 1974 
a/c 77:94 82°85 86:93 
d 6:976 | 7:3562 7614 


Here d = c(1 + nq) gives the distance of the start of the point-binomial from the 
centroid vertical. The three point-binomials are therefore 
77°94 (8881 + -1119)9%%, 
82°85 (8936 + °1064)"°*, 
86°93 (89985 + °10015)"°™, 
respectively. 
These three point-binomials are represented in Plate 8, fig. 3. It will be noticed 
that they all lie very close to the barometric curve; they would be still closer if that 
curve were a real curve and not a polygonal line. The total areas between bimomial- 
polygons and observation curves, treating all parts as positive, are for the three cases, 
10°3, 10°5, 11°0 sq. centims. respectively, or taking the base range to be 23 centims., we 
have mean deviations from the observation curve of ‘448, ‘457, 478 in the three cages 
respectively. Thus the method of trapezia gives slightly the best result ; the method 
of concentrating along ordinates the worst result. The total area of the curve being 
171°6, we have from another standpoint, mean percentage errors* in the ordinates 
of about 6°03, 6:06, and 6:3, respectively. The generalised probability curve, if fitted 
to the same observations, gives an areal deviation of 7 sq. centims., or a percentage 
error of about 4. Thus it is very nearly one-third as close again as the point-binomials. 
* The “ percentage error” in ordinate is, of course, only a rough test of the goodness of fit, but I have 
used it in default of a better. 
