504 
Proceedings of the Poyal Society of Edinburgh. [Sess. 
xfif + x 2 uf ] + x 3 u[ 3) = 2/i , 
X x U ( f + X 2 U { f + X z uf = 2/2 , 
X x U [ f + x 2 uf + x 3 uf = 2/3, 
and, 0 being Jacobi’s determinant of / x , / 2 , / 3 with respect to a; 1 , cc 2 , & 3 , 
there thus follows 
(a) X x (f> = 2/l(4 2) 4 3) - U‘f )u 3 ) ) + 2/ 2 (ttjM 3) - wf <>) + 2/ 3 «>i4 3 ) - «4 3) «4 2) ) , 
with similar expressions for x 2 f , x 3 ff ; so that any set of values of 
which makes / x , f 2 , / 3 vanish will make vanish also. The formal 
enunciation of this result is then given, and it is pointed out that the like 
theorem holds when there are n homogeneous functions all of n variables 
and of the r th degree. 
From (a) by differentiation there is next obtained 
dd> 
x ^ + + 
= 2 <(«> - wifW, 21 ) + 2a?'(af af - af af 3 ) + 2 <«>m!?> - 
+ 2/\ 4 («?“? - alaf) + 2/ 2 -— (afaf 1 - afaf 21 ) + 2/ s -^(afa!, 31 - afa!? ) , 
da?-, ^ “ 
dx. 
'dxy 
and thence 
~ 4 > 
02 y 
= 2/,— (wf?4 3) - 4 MC + 2/ 2 — (XX ~ XX) + 2/3 — « } z4 3) - XX ) , 
OXy OXy OXy 
so that any set of values of which makes the ternary quadrics j \ , 
f 2 , / 3 vanish, will make the first differential quotients of the determinant of 
fy , f 2 , / 3 vanish also, — a second theorem which is asserted to hold when 
the number of homogeneous functions is n, the number of variables n, 
and each function of the r th degree in the said variables. 
Combining the two results, and using later phraseology, we may say 
that When n n-ary 11 -tides vanish , their Jacobian and each of its first 
differential-quotients will vanish also. 
The connection of this with the problem of elimination can be indicated 
in a few words. Jacobi’s determinant <j> being of the third degree in x 1 ,x 2 , 
x 3 , its first differential-quotients are like fi, f 2 , / 3 linear in Xy , Xy , x 3 , 
x 2 x 3 , x 3 Xy , XyX 2 ; and consequently the resultant is at once obtained as a 
six-line determinant. 
