176 
Transactions of the Italian Society for 1804, he set forth 
these doubts, supporting his views by a reference to the results 
obtained by himself in his earlier researches. This work of 
Malfatti gave rise to two excellent Memoirs by Ruffini, also 
published in the Transactions of the Italian Society, in the 
first of which he disposes of Malfatti’s objections, and in the 
second he shows a priori that the quintic resolvent is of the 
sixth degree, and that each of its roots is of four dimensions 
with respect to the roots of the quintic. Sig. Brioschi in the 
Memoir above referred to, shows that the sextic equation 
calculated by Mr. Cockle (now His Honour The Chief Justice 
of Queensland) and myself, and which we communicated to 
this Society in 1858-59 (see Memoirs, vol. XV., pp. 131-142, 
and pp. 172-219), is substantially the same as that calculated 
by Malfatti. Sig. Brioschi further shows, in an Appendix 
to his Memoir, that Malfatti’s resolvent may be readily 
transformed into the sextic equation recently calculated by 
Mr. Cayley. {Phil. Trans, for 1861, pp. 263-276.) 
In a work entitled Die Anfiosung der Gleichungen funften 
Grades, by Adolf von der Schulenburg, Hauptmann, a D., 
published by H. W. Schmidt, Halle, 1861, I find calculated 
a certain sextic equation whose roots are of two dimensions 
with respect to the roots of the quintic. On comparing this 
sextic with Mr. Cayley’s, and making allowance for difierence 
of notation and for the circumstance that Herr Schulenburg’s 
sextic is calculated for the quintic wanting in its second term, 
whereas Mr. Cayley’s is for the complete quintic, I find that 
the two exactly coincide. By means of the seminvariant 
process I have completed the coefficients of Herr Schulen- 
burg’s sextic, that is to say, I have determined the form of 
those coefficients for the complete quintic, and I have thus 
verified Mr. Cayley’s result. 
Touching the theory of transcendental solution, I nmy 
state that my own efforts in this department are at present 
being directed to the determination of the most general forms 
