503 
1908-9.] Dr Muir on the Theory of Jacobians. 
correctly, to prove it, and to elucidate it by a commentary. The enuncia- 
tion is — If f 1 = a 1 , f|=a 2 , . . . . , f n = a„, where the as are constants , the 
functional determinant 
+ 
¥i ¥2 
d'Xj dx 2 
hfn 
dx„ 
will not be altered in value if before performing the differentiations 
every function 1] be transformed in any way whatever by means of the 
equations f i+1 = a i+1 , f i+2 — a i+2 f n = a n . On looking back it will 
be seen that Jacobi had previously not excluded the use of the first i— 1 
equations in making transformation of f . His proof now depends on 
taking the a’s to the same side as the f s ; denoting the resulting equations 
by (p 1 = 0 , <p 2 = 0 (p n = 0 ; applying his theorem ( De determ. fund. 
§ 10) to obtain the desired determinant in the form 
(-I)"' 
+ <tyl ¥2 
dx 1 dx 2 
dfn 
dx 
2 ± 
dy>, 0c j>< 
0a„, 
da [ 0a o 
and then showing that the denominator of this is ( — 1)' 
closes by assuming as allowable that 
4>i = hy f l + a iff 2 + . . . + hf f n , 
and having thus obtained 
His commentary 
2 ± 
d<f>i d<f> 2 
0$/-^ 0^2 
dfn 
dx„ 
2 • ■ • V)- 2( m 
dx ! dx 9 
¥ 
dx , 
he concludes that the condition for the equality of the two functional 
determinants is that the determinant of the As shall be equal to 1. 
Hesse (1844). 
[Ueber die Elimination der Variabeln aus drei algebraischen Gleich ungen 
voin zweiten Grade mit zwei Variabeln. Crelles Journ., xxviii. 
pp. 68-96: or Werke, pp. 89-122.] 
Having demonstrated that for the elimination of tfy-j. y C) y from the 
three quadrics f x , / 2 , f ?> it was important to discover a function of the 
third degree possessing certain properties, Hesse proceeds (§ 11) to show 
how such a function may be obtained. From a well-known theorem 
regarding the differentiation of homogeneous functions he has, on putting 
u (A) for , 
ox k 
