James Henderson 
175 
Suppose (p + n) — m : then a'- = ., , — — . 
^ 'm-{in +1) 
The coefficients a^, a^, ... etc. are given below: 
= 0, a., = 0, (Is = ^=2 ! a/ , 
V 3 ! ^ (//i, + 2) 
^4 = -7= 3 ! . „, \ — ^ {bm + o)K 
a. 
-V ^pg (,^ + 2)(m + 3)(m + 4)| pq V + > 
1 1 fmn'H+l)--^ 
a,i = -7= 5 ! ; 
V6~! + 2) + 3) (ni + 4) (m + 5) ( 
+ ^«l^JLll(„,2 _ g2/yi - 60) - f (2»i^ - 41m- - 154m - 120)}- , 
Vfi V ~pq (m+ 2) (7H + 3)(/>i + 4)(m + 5) (m + 6) 
(wi + 1)2 ('Ht + 1 ) -, ^ , , ^„ , 
- f\ (7™=* - .59m- - 342?H - 360} [ , 
1 1 
«8 = -7^ 7 ! 
VS ! ' (wi + 2) {m + 3) (m + 4) (m + ,5) (m + 6) + 7) 
m« (m + 1)3 _^ {47m-^ - 853m - 2100} 
+ — {- 251m3 + 1503»i^ + 9974ni + 10920} 
60pq ^ 
+ oV, {127bM^ - 1 6^7 - 44512m= - 104364^ - 65520} 
An additional coefficient was calculated for one of our examples, but it was not 
considered worth while woi'king it out algebraically. 
The coefficients in the tetrachoric expansion obtained by this latter method, 
that is, by using the property of tetrachoric functions as semi-orthogonal functions, 
are identical with those obtained from the first method, which consisted in equating 
moments of the functions on both sides of the equation. Thus we are led to the 
same expansion in both cases. 
(6) The numerical results are certainly interesting but from the utility point of 
view they are not very satisfactory. Tables I — VIII contain these results in a 
convenient form ; the values of the coefficients dg and a/, the tetrachoric functions, 
the successive terms (— ajT., and — a/xs) and the values of the series up to the term 
containing are given. It is to be noted that the coefficients do not appear in 
Tables II and IV but as these are the same as in Tables I and III respectively it 
