12 
in Analysis, which appeared in the Philosophical Transac- 
tions for 1844, part II. I found in fact that writing i° for 
x, and D for x~ or the differential resolvent of thetrino- 
’ dx r/0’ 
mial equation (I) may be made to take the form 
D(D-l)(D-2) . . . (D-»+2)y-(D-^)(D-®2=?) 
( D -i^). . ... (A) 
the only exception being the case n= 2, in which the resolvent 
contains a term independent of y. 
Using the ordinary factorial notation, that is to say, 
representing 
(w) (« — 1) (w — 2 ) . ( u — ?■+ 1) 
by \u\ r , the form (A) may be written 
n n ~ l 
n d 
-x- 
•1 dx 
2 n — 1 - | 
n— 1 J 
x n ~ l y— 0 . . . (B) 
In the Proceedings of this Society (vol. II., pp. 181 — 184) 
for the 4th of February last, I gave, without the details of 
calculation, the several differential resolvents for the succes- 
sive cases ji = 2, 8, 4, 5; and these results Mr. Rawson, of 
Portsmouth, has#indly verified. I gave, also, in the same 
Paper, the Boolian (symbolical) form of the resolvent for the 
biquadratic, and this seems to have suggested to Mr.. Cayley 
an investigation, in which he showed, by the aid of Lagrange’s 
theorem, that the equation (B) holds for all values of n. 1 
had the honour of communicating Mr. Cayley’s investigation to 
the Society on the ensuing 18th of February, and an abstract of 
it appeared at p. 193, vol. II., of the Proceedings. The Paper 
i 
itself will be printed in the forthcoming volume of Memoirs. 
Before receiving Mr. Cayley’s remarkable analysis, I had 
