1903 - 4 .] 
Dr Muir on General Determinants. 
63 
In modern phraseology its formal enunciation might stand as 
follows : — 
S being any axisymmetric determinant , It the determinant got by 
deleting the first row and first column of S, Y the determinant got 
by deleting the first row and second column of S, and Q the 
determinant got from It as R from S, then , if It = 0 
SQ = - Y2; 
and the theorem in general determinants whose validity is 
warranted by the proof given is in later notation — 
If | b 2 c 3 d 4 | = 0, then | a 2 c 3 d 4 | • | h 1 c 3 d 4 | = - | aib 2 c 3 d 4 | • | c 3 d 4 | . 
This, it is readily seen, is not a very obscure foreshadowing of 
Jacobi’s identity 
I AiB 2 1 = | VA | • I c 3 rf 4 | . 
Jacobi (1829). 
[Exercitatio algebraica circa discerptionem singularem frac- 
tionum, quae plures variables involvunt. Crelle’s Journ ., 
v. pp. 344-364.] 
In the ordinary expansion of (ax + by + cz - t)~ l there are 
evidently only negative powers of x and positive powers of y and 
z; in the like expansion of (b'y + cz + dx-t , )~ 1 there are only 
negative powers of y and positive powers of z and x; and 
similarly for (c"z + a"x + b"y - 1")- 1 . It follows from this that the 
ordinary expansion of (ax + by + cz - 1)- 1 . (b'y + cz + ax - t')~ l . 
(cz + ax + V'y-f)- 1 , looked at from the point of view of the 
powers of x, y , z, contains a considerable variety of terms ; for 
example, terms in which negative powers of x occur along with 
positive powers of y and z , terms in which x does not occur at all, 
and so forth. There is thus suggested the curious problem of 
partitioning the fraction 
1 
(ax + by + cz-t) (b’y + cz + ax - t') (cz + ax + U'y - 1") 
into a number of fractions each of which is the equivalent of the 
series of terms of one of those types. This is the problem with 
which Jacobi is here concerned. 
