90 Proceedings of the Eoyal Society of Edinburgh. [Sess. 
Let us consider now the differential equation for Legendre polynomials, 
(x 2 -l)y" + 2xy'-n(n+l)y = 0 .... (E,) 
and its equation of Didon for the order n, which is 
(x 2 - l)z" — 2(n - l)xz — 2nz = 0 .... (D 2 ) 
X 
In this last equation, let us replace x by -^=, and cause n to tend to infinity. 
We obtain 
z" + 2xz' + 2z = 0 (D 2 ) 
which equation is itself the equation of Didon for 
y" + 2xy’ + 2(n + 2)y = 0 .... 
But (E 2 ) is verified by 
(E 2 ) 
-h -* 
1 * e 
W 
n s 
9 2~5 
where the W function is of the type of the parabolic-cylinder functions. 
The connection between it and Legendre function is therefore established, 
through their equations of Didon. 
Let us come now to the field of two variables, and consider the poly- 
nomials V m , n {x, y ) studied by Hermite, which, as we mentioned in our 
Introduction, Appell showed to be hyperspherical zonal functions. The 
Y m> n function satisfies the system 
(Sj) 
^(1 - x l )r - xys - (n + 3 )xp + myq + m(m + n + 2)z — 0 
\(1 - y 2 )t - xys - (m + 3 )yq + nxp + n(m + n + 2 ) z = 0, 
of which the system of Didon for the orders m and n is 
(*i) 
f( 1 - x 2 )r - xys + (N - 3 )px + N qy + 2 N* = 0 
1(1 - y 2 )t - xys+ (N - 3 )qy + Rpa? + 2Nz = 0, 
y 
where N = m + n. Replacing in it x and y by -j == and -y==, and making 
then N infinite, we obtain the system 
(A 2 ) 
itself the system of Didon of 
(S 2 ) 
fr 4 - px + qy + 2z — 0 
[t+px + qy+ 2z = 0, 
cr + px + qy+(m + n + 2)z = 0 
1 1 + px + qy + (m + n + 2)z = 0. 
But a solution of (S 2 ) is 
~i ~i — i(sc 2 +2/ 2 ) tt- 
z — x y 6 m+w+i 
- 1 , -1 
