1901 - 2 .] Dr Muir on the Theory of Jacobians. 
159 
it may be fully described as the summation and development of a 
series of memoirs of which the former is the last. As is natural, 
therefore, six pages of it at the outset are occupied with an intro- 
duction, in which the main points of the said series of papers are 
recapitulated. Seven-and-twenty pages are then devoted to 
“ Problema I,” which may be roughly characterised in later 
phraseology as the problem of the linear transformation of an 
n - ary quadric. Then follows “ Problema IT,” the solution of which 
occupies pages-34-50. Its subject is the transformation of a very 
general multiple integral, and is closely connected with the subject 
of the preceding problem by reason of the fact that the integrand 
involves a power of an w-ary quadric. It is in this portion of the 
memoir that the special form of determinant which we are now 
considering makes its appearance. 
From two particular results in the previous papers referred to, 
Jacobi infers the existence of a general theorem which he states, 
adding that the demonstration is, however, not so easy, and that as 
a contribution towards it he will enunciate certain general theorems, 
the proofs of which for the sake of brevity are suppressed. His 
starting-point is the following : — 
If i Y , i - 2 , * * • , £»— 2 an V given functions of ^ , v 2 , . . . , v v _ 2 , 
From this he proceeds to cases where there are additional £ 
functions not independent of the others, enunciating and proving 
the result when there is one connecting equation, contenting 
himself with the mere enunciation for the case where there are 
two connecting equations, and leaving these to suggest the general 
theorem. The two enunciations are — 
Theorkma 1. Datis $ 1} $ 2i •••, 4-i ut functionibus 
ijpsarum v 1 , v 2 , . . . , v n _ x , si inter variabiles Mas datur 
aequatio 
then 
9£ 2 * * ■ d£?i- 2 = 
d h d A... d A=i 
dv 2 . . . dv n _ 2 . 
3iq dv 2 dv-n -2 
erit 
dfe • • • 
(X 
dv 2 . . . dv n _ 2 
