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XXXIV. Tables of the Sturmian Functions for Equations of the Second, Third, Fourth 
and Fifth Degrees. By Aethue Cayley, Esq., F.R.S. 
Received December 18, 1856, — Read January 8, 1857. 
The general expressions for the Sturmian functions in the form of determinants are at 
once deducible from the researches of Professor Sylvester in his early papers on the 
subject in the Philosophical Magazine, and in giving these expressions in the Memoir 
‘ Nouvelles Kecherches sur lesFonctions de M. Stuem,’ Liouville, t. xiii. p. 269 ( 1848 ), I 
was wrong in claiming for them any novelty. The expressions in the last-mentioned 
memoir admit of a modification by which then’ form is rendered somewhat more 
elegant ; I propose on the present occasion merely to give this modified form of the 
general expression, and to give the developed expressions of the functions in question 
for equations of the degrees two, three, four, and five. 
Consider in general the equation 
\J—{a,h,....j,kj3c,lf, 
and write 
V={a,b,...j 1)”“' 
Q=(^, ...j, kJj]c,lY-\ 
then supposing as usual that the first coefficient a is positive, and taking for shortness 
n„ n^, &c. to represent the binomial coefficients ^ — ?, &c, corresponding to 
the index [n — 1), the Sturmian functions, each with its proper sign, are as follows, viz. 
U, 
P, 
P, Q 
a, b 
xV, 
P, 
.rQ, 
Q 
a. 
• 
nfi. 
a. 
n,c. 
b 
nyc. 
nfi. 
n.f. 
n^c 
xF, 
P, 
x^Q, 
xQ, 
Q 
a. 
n 
b, 
•9 
n,b. 
a. 
•9 
n, c. 
b. 
• 
n^c. 
n,b. 
a. 
nfi. 
n, c, 
b 
n4. 
n^c. 
w, b, 
n^e. 
n^d. 
n^c 
n,e. 
n^ d, 
n.,c. 
n,f 
n^e. 
nfi 
where the terms containing the powers of x, which exceed the degrees of the several 
MDCCCLVII. 5 c 
