A. Ritchie- Scott 
413 
and the last term, i.e. when s = q, is 
z. 
'p\ ql 
Hence we may write 
p\q\^'^-'\p\ -"ql ip-l)l{q-l)lir{p-2)l{q-2)]2l ■••p 
If now we write <f>'' '\lr'J{z)= Tp^ q {x, y) . z, 
so that Tp^ q {x, y) might be termed a polychoric function in two variables or 
better a double polychoric function, we shall have 
TpA^,y)^Tp{^) Tq{y) Tp_,{x ) Tq_Ay) r Tp^,(x) Tq_,( y) _ 
p'lql p\ ' ql (i9-l)!'(?-l)!l! (jo-2)!'(g-2)!'2! 
J^'^ Tp_,(x) Tq_,{y) i-ry 
ip-s)r{q-s)l si 
p ox q being taken as the upper limit of s, whichever is the greater. When q — 0 
Tp,,(x,y )^Tpix) T^^Tpix) 
pi 01 pi ' 01 pi ^ ^' 
so that the single polychoric function is the particular case of a double polychoric 
function, viz. when the order with respect to one of the variables is zero. 
The above expansion may be reversed in a manner similar to the process in 
equations (54) to (58) giving the result 
Tpjx) Tq(y) ^ Tp^qjx, y) Tp_,, g -i {OS, y ) T p_^,q.,(x, y) 
pi ql p\ql {p-\)l{q-i)lir {p-2)l{q-2)V 2r ■■■ ^ 
7. Some properties of the Polychoric Functions. 
We have already seen (53) that 
^ 4- p ^"-^ + OQ^ +... 
nl ^^2(n-2)r-^ 2^2! (n-4)!^ 
X'' X X 
^nC^^'^h (n^^l 2^2!(/i-4)! ~ " 
.(71). 
Putting \ll—p = q and therefore p = 1 — we have 
^'2{n-2)r^ ^'2\2l{n-^)r 
_ x"- x^-'^q'- x'^-*q* _ 
~nl~ 2(w- 2')!"''22.2!(n-4)! ■■' 
T 
?(! 2(«-2)! 2^2!(«-4)! 
x\ 
.(72). 
