1901-2.] Dr Muir on the Theory of Jacobians. 
155 
in x~ l y~ x , the second equivalent to a series of terms consisting of 
negative powers of x and positive powers of y, and the third 
equivalent to a series of terms consisting of negative powers of y 
and positive powers of x , it is ciear that the three portions of the 
expansion of (ax + by)~ l (b'y + axf 1 have been isolated and 
summed. 
Now it will be noticed that a common factor of the three parts 
is the determinant | ah' j , or as Jacobi, following Cauchy, writes it 
(ah'). The corresponding factor in the next case, where there are 
three linear expressions and three variables, is found to be ( a b'c ") ; 
and Jacobi then makes a generalisation regarding the first of the 
partial fractions in each case, viz., to the effect that the coefficient 
of 
/y» l/y» I vi 1 'yt 1 
tAJ * * • V — 1 
in the expansion of 
u~ x ufuf .... u~\ 
i.e. of 
(ax + bx 1 + cx 2 + . . . ) _1 (b'x l + cx 2 + . . . . ) -1 (c",x 2 + .... ) _1 ... . 
is 
(a b'c". 
—the result being, so to speak, the discovery of the generating 
function of the reciprocal of a determinant. 
Shortly after this follows the passage which is interesting in the 
history of Jacobians. It stands as follows 
“At theorematis, de quibus in hac commentatione agimus 
et quorum modo mentionem injecimus, latissimam conciliare 
licet extensionem. Ponamus enim, u - t , u 1 — t' , . . . iam series 
esse quaslibet, sive finitas sive infinitas, ad dignitates integras 
positivas elementorum x , x Y , ... procedentes, quarum serierum 
t, t\ . . . sint termini constantes. Sint porro in seriebus illis 
u, u lt u 2 , ... termini, qui primas ipsorum x, x l9 x 2 , 
dignitates continent, respective ax , b'x 1 , c"x 2 , . . . , ac pon- 
amus, uti in casu lineari, fractiones (u — t)~ l , (u 1 - £') -1 , 
(u 2 — t")~ l , . . . evolvi respective ad dignitates descendentes 
terminorum ax, b'x Y , c"x 2 , .... Yocemus porro A deter- 
minantem differentialium partialium sequentium : 
