421 
1907-8.] Dr Muir on the Theory of Hessians. 
for all values of r and s * ; and consequently from (y) 
2 dx 0 
vXjdx-fc ( ox ^ 
+ 
= -{m- l)N»q j xf ^ 1 + xf- 
( 0X 1 3^2 
= ~{m- l)(m- 2)Nay%, 
3m* 
whence 
+ 
3 2 A 
- (?7i — l)(m - 2)N . 
d 2 u 
dxfix k 
This result we may formally enunciate as follows : — If the first differential- 
quotients of a homogeneous rational integral function all vanish, the 
elements of the Hessian of the function are proportional to the elements of 
the Hessian of the Hessian. 
Terquem, 0. (1851, March). 
[Note sur les determinants. Nouv. Annates de Math., x. pp. 124-131.] 
This is an elementary exposition of Hesse’s determinant, with simple 
illustrations from algebraic geometry, the property of “ covariance ” being 
made prominent. A curious distinction is made between what are called 
the “ first ” and “ second ” determinants : for example, 
4 ac — b 2 
is styled “le prewAer determinant de la fonction hexanome du second 
degre a deux variables,” and 
8 acf+ ibde - 2 ae 2 - 2 cd 2 - 2fb 2 
the “ second determinant de la meme fonction rendue homogene et ternaire.” 
The former is, in fact, the determinant of 
ax 2 + bxy + cy 2 + dxz + eyz +fz 2 
or ax 2 + bxy + cy 2 + dx + ey +f 
with respect to x, y ; and the latter the determinant with respect to x, y, z. 
* The first two of these results, which follow from (a) and (£), there is no pressing reason 
for mentioning : it would have been equally pertinent to note that u= 0. The third result 
Jacobi probably obtained (see Crelle’s Journal , xv. p. 304) by taking in every possible way 
n - 1 of the initiatory set of equations and deducing 
aq : $2 : : x n = Un : U21 • • • • • • tAi , 
= H] 2 • H22 • • • * • • U?i2 , 
= U lB : l T 2n :.•••: U nn . 
This implies that U rs : U rs / = x s : x s > 
and U S ' r : U s v = x r : x r < ; 
and from these, by reason of the equality of U rs / and U S ' r there is got by multiplication 
U rs : U s v = x s x r : x s > x r > . 
