20° THEORY OF STATISTICS. 
binomial distributions. It will have been noted that any one 
term—say the 7th—in one series is obtained by taking ¢ times the 
rth term together with p times the (r—1)th term of the preceding 
series. Now if AP, CR (figure 46) be two verticals, and a third, 
BQ), be erected between them, cutting PR in , so that 
AB :BC :q:p, then 
BQ=p.AP + q.CR. 
(This follows at once on joining AR and considering the two 
segments into which BQ is divided.) Consider then some 
binomial, say for the case p=1, g=2. Draw a series of verticals 
(the heavy verticals of fig. 47) at any convenient distance apart 
on Bpc 
Fre. 46. 
on a horizontal base line, and erect other verticals (the lighter 
verticals) dividing the distance between them in the ratio of 
q:p, viz. 3:1. Next, choosing a vertical scale, draw the binomial 
polygon for the simplest case n=1; in the diagram XN has been 
taken = 4096, and the polygon is abed, 0b = 3072, 1lc=1024, The 
polygons for higher values of » may now be constructed graphi- 
cally. Mark the points where ab, bc, cd respectively cut the 
intermediate verticals and project them horizontally to the right 
on to the thick verticals. This gives the polygon ad'c’d’e for 
n=2. Forob =gq.0b, 1c'=p.0b+q.1c, and so on. Similarly, if the 
points where a®’, b'c, etc.,, cut the intermediate verticals are 
projected horizontally on to the thick verticals, we have the 
polygon ab”¢"d"¢"f” for n=38. The process may be continued 
96