James Henderson 
179 
similar manner ; for, if we regard the graphs as a wave, it will be noticed that at 
first the amplitude of the wave is big, decreases gradually up to a term in the 
neighbourhood of T20 and thereafter increases more and more rapidly. This can be 
explained fairly easily ; as s increases the tetrachoric functions t^. do not increase or 
decrease steadily but vary in sign and remain of the same order of magnitude. The 
coefficients vary in much the same way (except that they are all positive) up to 
a certain point and then begin to increase very fast. In equation (xv) we had 
(Vsa, + \/(s- l )a. 
i.e. Ug+i is of order ^ - [ug + rts_2}, so that as s increases there comes a time when 
1 
^/s overcomes the reducing effect of and then the coefficients will continually 
•1630O 
•I6200 
•I610O 
•160001- 
15900 - 
•15800 
•15700 
• 15600 
15500 
I5400 
■15300 
19324- 
,-0. .. 
P Q 
•--er -o--- 
'■©•■' 
"1 — r 
I I I 
I I I I ~r 
'''2{l-a)% T2 T3 % % Tg Tg Ttj Tio T|| 1]2 T|j T|4, T|5 T|5 T,, Tjg Tjo T21 '^22'^23'^2A-'^Z5'^26'^27'^28'^2^'^30 
NUMBER OF TERMS 
■ Fig. 4. 
increase. For higher values of this turning point will not be arrived at so soon 
and the points will hang closer to the ' true value ' line for a greater number of 
terms, but it does not seem likely that the values of the series will tend to a definite 
limit. The equation for the modal expansion coefficients is a similar one and these 
coefficients behave in the same way. 
Turning our attention to the expansions from the mean, Fig. 1 (and Fig. 3 to 
a less extent) would seem to suggest that the tetrachoi'ic series gives quite a good 
approximation to the value of the integral. Although some of the points are very 
12—2 
