239 
for the quartic, it is 
( d -i)( d -4>-t) 
y D(D — 1)(D — 2) 
and for the quintic, it is 
9\/_ 13\/_ 17a 
e 3 « y =0; 
y- 
( D -D( D -f)( D -¥)(»-f) 
D(D — 1)(D — 2)(D — 3)' 
40 
6 y— o, 
which are all comprehended under the general form given 
on page 199. The induction, though incomplete, is yet 
sufficiently wide for the purposes of the present theory, inas- 
much as, when n is greater than 5, the given w-ic equation 
cannot in general be reduced to the trinomial form with which 
Ave are noAV Avorking. But Mr. Cayley’s brilliant piece of 
analysis, of Avhich an abstract is given on page 193, enables 
us, as avc have seen, to establish the theorem in all its 
generality. 
I noticed in my last communication the exceptional case 
n=2. The folloAving remarks on the same subject by Mr. 
Cayley will be read with interest. I had taken the liberty 
of calling his attention to the form of the resolvent for 
the quadratic, and replying in a letter to me, under date 
25th February, 1862, he says, “ I ought to have seen, 
and after Avriting the note on the Differential Equation did, 
in fact, see from your paper that n— 2 is an exceptional 
case : the d priori reason is, that for the equation 
y— u +\y~i or y 2 — 2y-F u=0. 
Ave have (not as in the general case yi + ^'k&c. =0) but 
y x + y 2 — 0. Hence for n— 2, the differential equation cannot 
be of the form 
for if so, its solution, Avhich would be of the form y— CY, 
