James Henderson 
161 
(2) It is well known that a wide range of frequency distributions can be 
adequately represented by one or other of the curves 
y = y/„e-v.'-/«(i + 
a 
(vii). 
By a change of origin and the appropriate stretch or squeeze these may be 
reduced to 
y = y^x>^'~^e~'^ .....(a)! 
y (vii) bis. 
and 7/ = ^„.*'™ -1 (1 - xY'-^ {}))] 
Now, generally, it is not the ordinates of these curves which are required but 
the areas of certain portions, or in other words the probability integrals of these 
skew curves. The total range for (vii) bis(o) is 0 to oo and for (h) is 0 to 1 ; since 
/. 00 
/ n 
and A-™' "Ml - *')"'^ ' ^ dx = B ( m, , ia.^, 
we may take these probability integrals to be 
1 
I(p,v) = „^ v''-'e-"dv 
1 r*' 
and B(v, m,, 7n..) = ~ \ v"'-Ul- vy''-'^dv, 
B(?».j, ??io) jo 
which are the ratios of the incomplete to the complete V- ami B-functions. 
The equations on p. 158 show us that if either of the frequency functions (vii) is 
expressible in a series of tetrachoric functions their probability integrals (assuming 
convergence) will also be. Now there is no doubt that a large mass of material does 
not differ practically from the forms in (vii) and accordingly if the above probability 
integrals cannot be adequately expressed in a series of tetrachoric functions, we 
may be certain that tetrachoric functions do not furnish a suitable method of 
representing skew frequency. Accordingly our problem reduces itself to the 
following one : Can / (j:), v) and B {v, m^, m.^), or the Incomplete F- and B-functions, 
be represented with adequate convergency by a series of tetrachoric functions ? 
After examination of the numerical and gi'aphical results obtained, we are obliged 
to conclude that the answer to this question is in the negative. 
(3) Let us first consider the expansion in tetrachoric functions of the function 
y = xP-' e-^7r(jL)) (viii). 
In expanding this expression thei'e are at least two methods, which we ought 
to corjsider, and one may have advantages over the other as far as convergency is 
Biometrika xiv 11 
