A. Ritchie-Scott 
415 
Various functions may be expanded in terms of T. Thus since 
-e-^'^'(^o + ^i + ^2+ •••) (79). 
.-. 6'„ + ^, + ^,+ ...=e-4(-'^-i)' + i-'^' = e-^-^ (80). 
.-. e^^e^{e, + e, + d, + ...) (81). 
8. Expansion of the Polychoric Function. 
If we write Tp, Tq for Tp(x) and Tg{y) respectively we have seen that (1) Tp 
may be written Tp„, (2) Tq as Taq. The same symbol T may be used for the single 
or the double function, as its order is shown by the number of suffixes. 
p\q\~ plql ~ {p - l)!(g- 1)! ^ 2] " {p-2)f(^^^. ~ 3l " (p-S)l{q - 3)! 
yp^g _ / 1 ^0-1 T g_i ^ y 9-2 , ^ Tp-3 T ,]-s 
plql J \{p-l)l{q-l)l {p-2)l(q-2y. 2\'{p-S)l{q-S)\ 
■^p'^q [ ^i'— 1,9— 1 
plql Jip-l)liq-l)l^' ^^2), 
and Tpq=TpTq-p.q ^Tp_,^q_,dr (83), 
which gives a simple method of expanding Tyq. 
It may be convenient on occasion to express the above in the 6 notation 
epq = epe; -\ep_,,q_,dr (84). 
Having expanded Tpq we can at once write down the expansion (f)i'-\jri. The two 
following tables give the expansions required, viz. <f)i'yjri in terms of powers of x and 
y multiplied by z and its converse. 
TABLE I. 
xy products of z in terms of polychoric functions. 
1. xz = ^ {z) 
2. .r"z ={(f>"- + l}z 
xyz = {^yjr +r) z 
3. x'z =(</)^+30)2 
x~yz = {(f)'-^ + i/r + 2r<f>) z 
