64 Proceedings of Royal Society of Edinburgh. [sess. 
In the case of two variables he counts three types of terms, 
viz., that in which the indices of both x and y are negative, that 
in which the index of x only is negative, and that in which the 
index of y only is negative. In the case of three variables he 
counts seven types, viz., that in which the indices of x , y, z are all 
negative, the three in which the index of only one variable is 
negative, and the three in which the index of only one variable is 
not negative. These two cases are gone fully into, with the result 
that the expressions for the three aggregates in the former are all 
found to contain the factor {ab')~ 1 , and the expressions for the 
seven aggregates in the latter the factor ( a b'c")- 1 . The reciprocal 
of each of those factors is recognised as the common denominator of 
the values of the unknowns in a set of linear equations, a 
denominator “ quam quibusdam determinantem nuncupamus et 
designemus per A.” Its persistent appearance in the problem 
under discussion, — a persistency, in fact, sufficient to suggest the 
change of the numerator of the given fraction from 1 to [a b’) in 
the case of two variables and from 1 to (a b'c") in the case of three, 
— is remarked upon : — “ Quam determinantem in hac quaestione 
magnas partes agere videbimus, videlicet omnes Mas series infinitas , 
quas ut coefficientes producti propositi evoluti invenimus , ex 
evolutione dignitatum negativarum deter minantis provenire” Then 
fixing the attention on a unique term of the expansion Jacobi 
ventures on the generalisation that the coefficient of 
(x x 1 x 2 x n —f 
in the expansion of 
that is to say, of 
{ax + by + cz + . . ,)- 1 (b'y + cz + . . . )~ 1 {c"z+.... ) _1 
is the determinant 
(a b'c" )-\ 
No proof, however, is given, save for the cases where n — 2 and 
n = 3. The proposition is most noteworthy in that it supplies the 
generating function of the reciprocal of a determinant. 
To obtain a generalisation in a different direction, viz., from 
{ax + by) ~ 1 {bpy + ape) ~ 1 to {ax + by) ~ m (bgy + ape) ~ n , Jacobi pro 
