A. Ritchie-Scott 
409 
Any cf) integral may therefore be reduced one degree and since wo can evaluate 
f 
I z{x, h) d.r the formal solution of the problem is complete. 
J —00 
6. PolycJioric Functions. 
I shall now consider in detail the case for two variables, but before doing so 
it will be necessary to establish certain ancillary formulae wliich will greatly 
simplify both the process of integration and the computation when the formulae 
are used in numerical work. The first of these formulae are closely related to 
the well-known tetrachoric functions which have been tabulated in Tables for 
Statisticians and Biometricians (Cambridge, 1914), and to emphasise this relation 
I have ventured to call them polychoric functions. They are however already 
known as parabolic cylinder functions* (see Whittaker and Watson, Modern 
Analysis, p. 341, 2nd edition, Cambridge, 1915). Only the properties germane to 
the present issue will be dealt with here. 
From the definition of / {x', y) we have 
and we may write 
^, z' {x , y') = - ^ r^]z 
^jz {x , y) = ~ - — [•^-r—]z 
dy 
</>' (/) 
.(40), 
/ i\ X , 
''""^dx'-^ 
'dy') 
.(41), 
where ^' and -v/r' are functional operators. 
In most cases there will be gain in clearness and no loss in generality in 
writing x for — and y for — and when desirable the following contracted forms 
will be used : 
Since 
i>iz)=-[ i^+,.|)(.)=...; 
<l){z)=x.z, ylr{z)=y.z-^ 
.(42). 
(f)(y) = -r, iriy) = 
1 
.(43), 
[* For the history of the subject and applications of the functions; see Pearson, K., Phil. Tnins. 
Vol. 195, A, pp. 1-47, 1900; Whittaker, E. T., Proc. Loud. Math. Soc. Vol. xxxv. pp. 417-427, lit03; 
Pearson, K., "A Mathematical Theory of Random Migration," Drapers' Research Memoirs, Biometric 
Series, nr., Camb. Univ. Press, 1906; and Cunningham, E., Proc. li. Soc. Vol. 81, A, pp. 310-331, 
1908. Ed.] 
