[ 751 ] 
XV. On a Class of Invariants * 
By John C. Malet, M.A., Professor of Mathematics, Queen’s College, Cork. 
Communicated by Professor Cayley, LL.D., F.R.S. 
Received December 14,—Read December 22, 1881. 
I have not seen it noticed by any mathematician that in the theory of Linear 
Differential Equations there are two important classes of functions of the coefficients 
which have remarkable analogies to the invariants of Algebraic Binary Quantics; 
consequently I venture to call attention to their existence and also to give examples 
of their application in the present paper. 
For convenience I write the equation with binomial coefficients thus 
P+nP SA- 
dx n 1 dx n ~ 1 
n.n— 1..~ d n ~hi 
—nr~ Po J 
^dx n ~ % 
+R n y =0 
0) 
where of course p,. p* &c., are functions of x only. 
If now we remove the coefficient of ~~ by changing y to ye~n dx the equation, 
wanting the second term, may be written, 
where we have 
dry n.n—1. d n ~ 2 y ( n.n—l.n—%~d n ~ z y 
dx n ^ 1jd ^dar-s 
=o _ 
|4 
H = P ^- P 1 2 -^ 
G=2r 1 3_ 8 r 1 r 2 +r 3 -5 
I=-6P 1 «+12Pi'P,-4P 1 P,-8P,»+P*-^ 
( 2 ) 
* Since tbe publication of tbe abstract of this paper tbe Rev. R. Harley has mentioned to me that 
the first class of functions treated of here have been already investigated by Sir James Cockle ; having 
consulted the memoirs I was referred to by Mr. Harley, I think little similarity will be found between 
Sir James Cockle’s results and mine.— J. C. M. 
5 D 
MDCCOLXXXII. 
