151] 
471 
151. 
TABLES OF THE STUBMIAN FUNCTIONS FOB EQUATIONS OF 
THE SECOND, THIBD, FOUBTH, AND FIFTH DEGBEES. 
[From the Philosophical Transactions of the Royal Society of London, vol. cxlyii. for 
the year 1857, pp. 733—736. Received December 18, 1856,—Bead 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 Recherches sur les Fonctions de M. Sturm/ Liouville, t. xm. p. 269 
(1848), [65], I was wrong in claiming for them any novelty. The expressions in the 
last-mentioned memoir admit of a modification by which their form is rendered some 
what 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 
U = (a, b, ... j, k\x, l) n , 
and write 
P = (a, b, ... j\x, l)“ -1 , 
Q =(b, ... j, kjx, l) n_1 , 
then supposing as usual that the first coefficient a is positive, and taking for shortness 
n 1} n 2 , &c. to represent the binomial coefficients n —, -—y—^ > & c - corresponding 
to the index {n— 1), the Sturmian functions, each with its proper sign, are as 
follows, viz.