593 
NOTES AND EEFEEENCES. 
384. The conclusion arrived at Nos. 27—30 that the transformed curve of the 
order D +1 depends upon 4D — 6 parameters is at variance with Riemann’s theorem 
according to which the number of parameters is 3p — 3, (p Riemann =D Cayley), = 3D —3, 
and this last is the correct value. My erroneous conclusion is referred to in the 
preface to Clebsch and Gordan’s Theorie der Abel’sehen Functionen (Leipzig, 1866), 
“ Unter den von Riemann behandelten Theilen der Theorie haben wir die Frage nach 
der Anzahl der Moduln einer Klasse von Abel’schen Functionen ausschliessen zu müssen 
geglaubt. Diese Frage ist durch die scharfsinnigen Betrachtungen des Herrn Cayley 
Gegenstand der Controverse geworden: sie ist überhaupt wohl zunächst nur. durch tiefe 
algebraische Untersuchungen endgültig zu entscheiden, für deren Schwierigkeiten die gegen 
wärtig bekannten Methoden nicht mehr auszureichen scheinen.” In the case D (or p) = 3, 
my value is 10, Riemann’s is 9: that the latter is correct was shown by a direct 
proof in the paper Brill, “Note bezüglich der Zahl der Moduln einer Klasse von 
algebraischen Gleichungen,” Math. Ann., t. I. (1869), pp. 401—406 : the explanation of 
my error is given in the paper, Cayley, “ Note on the Theory of Invariants,” Math. 
Ann., t. hi. (1871), pp. 268—271. 
400. The question here considered, viz., the expression of a binary sextic f in 
the form v 2 — u 3 , v and u a cubic and a quadric respectively, forms the basis of 
the very interesting investigations contained in the Memoir, Clebsch “ Zur Theorie 
der binären Formen sechster Ordnung und zur Dreitheilung der hyperelliptischen 
Functionen,” Gott. Abh., t. xiv. (1869), pp. 1—59. Considering / as a given sextic it is 
remarked that the number of solutions, or what is the same thing the number of 
the functions u or v, although at first sight = 45, is really = 40 ; supposing that there 
is a given solution u, v, or that the sextic function is in the first instance given in 
the form v 2 — u 3 , then if any other solution is v!, v', we have v 2 — u 3 = v' 2 — u! 3 , where 
v', u' are functions to be determined: there are in all 39 solutions, a set of 27 and a set 
of 12 solutions: viz. writing the equation in the form (v+v')(v—v')=(u—u')(u—eu'){u—e 2 u'), 
e an imaginary cube root of unity, then either the v + v' and the v — v’ contain each 
of them as a factor one of the quadric functions u — u', u — eu, u — e 2 u (which gives 
the set of 27 solutions) or else the v + v' and the v — v' are each of them the product 
of three linear factors of the quadric functions respectively (which gives the set of 12 
c. VI. 75