72 
Proceedings of Royal Society of Edinburgh. [sess. 
and 
[y iY ,..yJ 
1 
Vdh • • ' Vr, 
(m) 
so that there is obtained 
as was expected. 
Jacobi (1834). 
[Dato systeniate n aequationum linearium inter n incognitas, 
valores incognitarum per integralia deiinita (n — 1) tuplicia 
exhibentur. Crelle’s Journ ., xiv. pp. 51-55.] 
This short paper is, as it were, a by-product of the investigation 
which resulted in Jacobi’s long memoir of the preceding year. 
Its only interest for us at present lies in the fact that values 
which are ordinarily expressed by means of determinants are here 
given in the form of definite multiple integrals. Indeed, instead 
of viewing the result obtained as being the solution of a set of 
simultaneous linear equations, it might be equally appropriate to 
consider the investigation as belonging to the subject of definite 
integration. It will suffice, therefore, merely to give a statement 
of the theorem arrived at. In Jacobi’s own words, it is, — 
“Sit propositum inter n incognitas z x , z 2 , . . . , z n systema n 
aequationum linearium 
b\\ z \ + b l 2 z 2 4 - • • • ■ + b ln z n = m 1 , 
b 2 l z 1 + b 22 z 2 +••••+ b 2 n z n = m 2 , 
b n iZ i + b n 2 z 2 + + b nn z n = m n ; 
statuamus 
X = \b n x x + &2i^2 + • • • + b n i&J 2 
+ \b l 2 x x + b 22 x 2 + • • • + b n 2 x n J 
+ \b ln x x + b 2 n x 2 + • • • + b nn xff , 
