ALGEBRAIC FUNCTIONS WITH AUTOMORPHIC FUNCTIONS. 
17 
where P (2) is a polynomial in 2, and a’s are constants. Now at 2 = 00 we must 
have a zero of at least the orcler-y. Hence P (2) = 0 ; and since near 2 = 00, R (z) 
can be expanded in the form 
n+2 / 
Pv (z) — -/e S ( ,2 4 - -f + . ■ 
r=i \ ^ 
r=l 
\ \ I I I 1 \ 
by equating to zero the coefficients of—, —, and —, respectively, we obtain 
n+2 
N a,.. — 
r=l 
0, 
n+2 
V 
r=l 
a+2 
V 
)■=! 
cL.e, = — 
3 (n +^) 
^ 16 ^’ 
a,,ey = 
a+2 
— A V f 
r=l 
These conditions enable us to write R (2) in the form 
p _ .s 1 , 3 - (^ + ^ + C/" ^ + . . . + 
(Z) — 1 tj X2 -Tie 
r=l(« - CT 
(2 - Cl) (s - C2) . . . (2 - e„+2) 
where Ci, c.2, . . . c,i_i are constants as yet undetermined. 
Hence the requh'ed analytical relation beftveen t and 2 is 
'll d* 2 
JL 5 / 'y\ - N' - -I- 
{n + 2) 2” + • 2” ^ + Ci2’' ^ + ... + c„_i 
(2 - Cl) (2 - Cg) ... (2 - e„+2) 
It will be seen that this is the differential equation for the quotient of two 
solutions of a linear differential equation of the second order with (n + 2) singu¬ 
larities, at each of which the exponent-difference is Such linear differential 
equations have been studied by Klein,"^ as being the generalisation of Lame’s 
equation; and Bocher’s book, ‘ Ueber die Reihenentwickelungen der Potential- 
Theorie’ (Leipsic, Teubner, 1894 ), is chiefly concerned with them. Bocher proves 
that the differential equations of harmonic analysis are limiting cases of them. 
We can trayisform this equation to a simpler form. 
Put 
C dz 
~~ J v/(2 - Cl) (2 — c,) ... (2 — c^+a) ’ 
w 
so w is a known function of 2. 
VOL. CXCII. 
* ‘ Gottinger Nachricliten,’ 1890, pp. 85-95. 
D 
