154 Proceedings of Royal Society of Edinburgh. [sess. 
minants to geometry ” which are often supposed to belong to a 
much later date. 
Jacobi (1829). 
[Exercitatio algebraica circa discerptionem singularem 
fractionum, quae plures variabiles involvunt. Crellds Journ.,, 
v. pp. 344-364 : also abstract in Nouv. Annates de Math., iv. 
pp. 533-535.] 
To the great mathematician whose name was ultimately associ- 
ated with determinants of this special form, they first appeared in 
a totally different connection. He was considering a problem of 
the partition of a fraction with composite denominator into others 
whose denominators are factors of the original, and the paper to 
which we have come concerns given fractions of the form 
{ax + by - ty 1 ( b'y + ax - t'y 1 , 
{ax -\-by-\-cz — ty 1 (b'y + c'z + ax - t'y 1 (cz + ax + b"y - fy 1 , 
The expansions of these clearly contain a variety of terms, the 
reciprocal of each linear expression contributing negative powers 
of its first term and positive powers of the others; and the 
‘ discerptio singularis 5 consists in obtaining fractions which produce, 
each of them, the aggregate of the terms of a particular type found 
in the expansion. Thus, to take the simplest example, viz., 
(ax + by)~ l (b'y -{-ax )- 1 , 
it is seen that the expansion of ( ax + by )~ 1 will contain one term 
with negative power of x. and others with a negative power of x 
and a positive power of y, that the reverse will be the case in the 
expansion of (b'y + ax)- 1 , and that the product of the two expan- 
sions will therefore contain a term in xr 1 y~ x , a series of terms with 
negative powers of x and positive powers of y, and a series of 
terms with positive powers of x and negative powers of y. How 
Jacobi establishes the identity 
(ax + by) 1 (b'y 4- ax) 1 
b 
ax -f by 
1 a! \ 
y b'y + ax) y 
where on the right there are three parts ; and as the first is a term 
