19 
In my third communication to the Society, entitled e< On 
the Theory of the Transcendental Solution of Algebraic 
Equations” (seepp. 237 — 241 of Vol. II. of the Proceedings ) , 
I gave Mr. W. H. L. llussell’s solution by definite integrals 
of a certain quartic resolvent, and in connexion therewith I 
remarked that Professor Boole had pointed out to me a method 
of solution which did not differ essentially from Mr. Russell’s. 
I have since found that there is an essential difference, 
inasmuch as Professor Boole’s method gives the solution in 
forms involving single, not (as in Mr, Russell’s solution) 
triple integrals. 
Soon after the publication of the Paper above quoted, I 
received from Mr. Cayley a letter (dated April 29, 1862,) 
containing some remarks on the subject, Avhich I have pleasure 
in appending to this communication. 
Mr. Cayley to Mr. Harley. “ The series for the solution 
of your resolvent equation should I think be assimilated to 
the form of the hypergeometric series. If instead of the 
ordinary notation 
F(a,/V/,*) = l + ^*+ 
a.a-\- 1 ./3./3-f- 1 
1 . 2 . 7 . 7 + 1 
x 2 -p &C., 
we write 
then naturally 
will denote 
IT 
a . /3 . 7 a . a-\- 1 . /3 . /3-p 1 . 7 . 7 T 1 
-.r+ 
r 2 J- 
1.2.a.S+l.e.e+l ^ 
and Mr. Russell’s series, writing them under the forms 
12 
2_ — 1 
12 ' 12 
9 
‘■r 
1 
X s + 
M2 12 / M2 12 / M2 12 ) 
1- 2- 
5_ 
T 
o 
1 
3 
