1907-8.] Dr Muir on the Theory of Hessians. 
413 
XXVIII. — The Theory of Hessians in the Historical Order of 
Development up to 1860. By Thomas Muir, LL.D. 
(MS. received February 24, 1908. Read March 16, 1908.) 
Special cases of the determinant 
d 2 u 
d 2 u 
d 2 u 
0^2 
dxdy 
dxdz 
d 2 u 
d 2 u 
d 2 u 
dydx 
di 2 
dydz 
d 2 u 
d 2 U 
d 2 u 
dzdx 
dzdy 
te 2 
where u is a function of x, y, z, . . . , may well have appeared at a very 
early date in the history of determinants. The case where u = ax 2 -\-2 bxy 
-f cy 2 may be viewed as traceable to Lagrange (1773), and the case where 
u = ax 2 -f- by 2 + cz 2 + 2dyz + 2ezx -j- 2fxy to Gauss (1801); but it is certain 
that in those cases the elements of the determinants were not looked on 
as second differential-quotients of u. The general conception first occurred 
to Hesse in the year 1843. 
Hesse, 0. (1844, January). 
[Ueber die Elimination der Variabeln aus drei algebraischen Gleich- 
ungen vom zweiten Grade mit zwei Variabeln. Crelles Journal „ 
xxviii. pp. 68-96 : or Werke, pp. 89-122.] 
In § 15 (p. 83) Hesse passes from the direct subject of his paper to the 
special case in which the three functions f v / 2 , / 3 are the first differential- 
quotients of the homogeneous function of the third degree 
or/ say, 
where each of the suffixes k, /x may be 1 or 2 or 3. The determinant, 
afterwards called the Jacobian , of f v / 2 , / 3 he says may in that case be 
styled “ the determinant of /.” This expression at once recalls that used 
by Gauss in 1801, namely, “ determinant of a form of the second degree/’ 
the determinant of ax 2 + 2bxy + cy 2 , according to Gauss, being b 2 — ac,. 
and the determinant of a^x 2 + a^y 2 + a z z 2 + 2b x yz + 2b 2 zx + 2b 3 xy being 
« 1 5 1 2 + a 2 b 2 2 -f- a 3 5 3 2 — a 1 a 2 a s — 2b 1 b 2 b s . The two usages, when Hesse’s is re- 
