84 
Proceedings of the Royal Society of Edinburgh. [Sess. 
Let us now divide x by m and y by n, causing m and n to tend 
simultaneously to infinity : we observe then that 
A (- y A 
li m ' n \m ’ n ) 
».»=« (y, m )(y'j n) 
e- 2 ^%(y + 7-8; y, y ; a, y), 
showing a connection between A Wj n and of the same nature as the 
connection between Legendre and Bessel functions. 
2. The Two-variable Polynomials n of Hermite. — These functions, 
introduced by Hermite,* arise from the derivation of an exponential 
where the exponent is a quadratic form of x and y ; their definition is 
0m+n 
- y ) 
where 
dx m dy n 
cf>(x, y) — ax 2 + Tbxy + cy 1 . 
It may be shown that this polynomial depends essentially on the function 
W m+n-l 
2 » ' ?» ?• 
3. The Two-variable Bessel Function of order Zero. — Several results 
have been published lately on the subject of new functions of two vari- 
ables possessing certain properties analogous to Bessel functions.*)* These 
two-variable Bessel functions are defined by the expansion 
y) u » 
or by the integral 
1 f n 
J n (x , y) = - I cos ( nu — x sin u - y sin 2 u)du. 
*} o 
It may be shown that the simplest of these functions, J 0 (x, y), satisfies the 
same differential equation as our solution 
e ~ iy %(h b - i^ 2 )- 
* CEuvres , ii, p. 293. 
f Of. a paper by Jekliowsky, with a bibliography of the subject, in Bull. Astron ., 
t. xxxv, 1918. 
