418 
Proceedings of the Royal Society of Edinburgh. [Sess. 
Hesse, 0. (1847, August.) 
[Ueber Curven dritter Classe und Curven dritter Ordnung. Crelle’s 
Journal , xxxviii. pp. 241—256 : or Werke, pp. 193—210.] 
Any homogeneous function of the m th degree in the variables x v x 2 , x 3 
being denoted by u, its first differential-quotients by u v u 2 , u 3 , and its 
second differential-quotients by u lv u 12 , . . . there is obtained from Euler 
^11^1 1” U 12*^2 4 U \Z X 3 = i. m ~~ f) W l > 
u 2l x x + u 22 x 2 + u 2 3 x 3 = (in - 1 )u 2 , 
u 31 x a + u 32 x 2 + u 33 x 3 — (m — \)u 3 , 
and thence, on solving, 
— = HnWi + UigWg + Uis'Wg , 
— X 2 — 4-1 21^^1 "1" 1^22^2 1" ^1“>3^3 J * 
m _ ^3 = Usfifi 4 ll32 M 2 4 Hgg^g , 
where, evidently, A is used for Hesse’s determinant of u, and U rs for the 
cofactor of u rs in A. Using in connection with the latter three equations 
the multipliers u v u 2 , u 3 and adding, Hesse derives the interesting result 
m 
m — UjjWj 2 + U 22 W 2 " + Ugg^g" + 2U 2 3W2 W 3 4 2U 31 w 3 ?q -f- 2X12^2^1^2 > 
and this by a process of differentiation leads to six results of the type 
^22^2.% + n l3 U Y U 2 — U 23 U p — 
The rest of the paper is geometrical. 
THy T-T- OCcfiCo a 
U 23»-7z ; 2 ^2 A - 
m — 1 " (ro- 1) 1 
Hesse, 0. (1849, January). 
[Transformation einer beliebigen gegebenen homogenen Function 
4ten Grades von zwei Yariabeln C relies Journal, xli. 
pp. 243-263 : or Werke, pp. 223-246.] 
A binary quartic u 24 being the only other homogeneous integral function 
whose determinant, in Hesse’s sense, is of the same degree as the function, 
there was naturally an inclination to make a study of its properties in the 
same fashion as had been followed with the ternary cubic. Analogous. 
