TRANSACTIONS OF SECTION A. 639 
sums. Next, three shows and counts of hands are to be made: (1) for less than 
A; (2) for more than A, but less than B; (3) for more than B, The results are 
a per cent. vote for less than A ; 100—a@ vote for more than A, 
6 per cent. vote for less than B; 100-0 vote for more than B. 
Numerous analogies amply justify the assumption that the estimates will be 
distributed on either side of their (unknown) median, m, with an (unknown) 
quartile, g, in approximate accordance with the normal Jaw of frequency of error. 
The following table of centiles (a better word than ‘ Per-centiles,’ which I originally 
used), having a quartile = 1, is founded upon that law. It is extracted from my 
‘Natural Inheritance’ (Macmillan, 1889, p. 205) to serve the present purpose. 
Centiles to the Grades 0°—100°. 
| Grades | 0° BEEN iFPers he eh Syhl VeAPOO Ne wBPL ath gexOPne fo MP oP APL da 
| | 
10° |—1:90 —182 |—1:74|—1:67 |—1:60 —1:54 —1:47|—1:42 |—1°36 |—1-°30 
0° = | —inf: | 3-48 |— 3-05 | 2:79 | 2-60 2-44 | - 2-31 |—2-19 |—2-c8 |—1-99 
20° aoa -- 1:20 |—1-15 |—1-10 —1-05 |—1-00 |—0-95 | —0-91 |--0-86 | —0:82 
| 30° —0'78 —0-74 —0°69 | —0°65 —0°61 —0°57 |—0°53 |—0-49 |—0-45 |—0-41 
40° | 0:38 |—0-34 |— 0:30 | — 0°26 | — 0-22 |—0-19 |— 0-15 |—0-11,| 0-07 |—004 | 
50° 0:00 | + 0-04 | +007} +011 +015 +019 +022 +0:26 +030) +0°34 
60° /+0°38 |+ 0-41 +0°45 | +049 | +0°53 +057 | +0°61 |+0°65|+0°69'+0°74 
a cO* '+0°78  +0-82 + 0°86 | + 0-91 | + 0°95 +1:00 +1:05 |} +1:10|+1:15 +1:20) 
| g0° | +1:25 |+1:30 | +1:36 | +1-42 |+1:47 4+1:54/4+1:60/4+1°67/+1-74 +182 | 
Ie 902 | +190 | + 1-99 | +208 |4+2:19/+2:31 +2:44|+2°60!+42-79 |+3-05 | +3:45 | 
| | ! I | | | 
Let a be the tabular number znelustve of its sign, that corresponds to the 
grade a°, and let 8 be that which corresponds to 6°, then 
m+qa=A; m+q3=Bb, 
whence 
ss 
m=-A~—a es =B-B LS ees 
B-a | B-a J 
Example :—A =100, B=500; a=40°, 6=80°, whence a= —0'38, B= +1:25 and 
m=193. The truth of the determination of 7 may now, if so desired, be tested 
by putting two new values A’ B’ to the vote, in the same way as A and B, but 
A’ and B’ should not differ much from m, and it should be an honourable under- 
standing that no member should deviate from his first opinion in giving his 
second vote. 
When about to utilise this method, A and B ought to be so selected that A 
shall secure not less than 5 per cent. of the votes, and B not more than 95, 
because the curve of error ceases to be trustworthy near to its extremities, but a 
dependence upon it within the limits of 5° and 95° will seem pedantic only to 
those who are unfamiliar with its nature and with its numerous and successful 
applications. 
It will be easily understood that this method is a particular case of the more 
general problem, that in any system of normal variables which has been arrayed 
between the grades of 0° and 100°, if the values be given that correspond to any two 
specified grades, those that correspond to each and every other grade can be found. 
I heartily wish that when occasion offers, some Assembly may be disposed to 
experiment on the above method. The calculators should, of course, rehearse the 
as beforehand, and be well prepared to carry it through both rapidly and 
surely. 
It is worth mentioning that when the above table is not at hand, a graphical 
substitute for it, that ranges between 5° and 95° and is true to the first place of 
decimals, may be quickly made by those who can recollect three simple factors. 
Thus, draw between two vertical limits, 0° and 100°, a straight line on squarely 
ruled paper, having a quartile equal to 1. Accept this line in lieu of the curve 
between 30° and 70°, add one-twentieth to the lengths of the centiles at 20° and 80°, 
