232 Proceedings of Royal Society of Edinburgh. [sess. 
It will be at once observed that Cauchy’s italic letters S, a, b are 
simply changed into Greek a, (3. 
The next sentence is : — 
“ Eadem notatione adhibita, sit 
B = 3 ± P1P2' • » • • Pn n J 
ubi ipsam B e quantitatibus /? fc (m) eodem modo compositam 
accipimus, quo A ex ipsis a k (m) componitur. Quibus statutis, 
observo fieri : 
B = A w -\ 
ac generalius : 
2 ± A'ft" = A-*a ± O • • • • <£’ •” (xx. 3) 
As for the first theorem thus formulated, the credit of it is, of 
course, due to Cauchy : the second, however, is new, being indeed 
the theorem referred to above under Minding as having been fore- 
shadowed by Lagrange, and left for over fifty years undisturbed. 
Jacobi evidently knew it in all its generality, for he adds — 
“ De qua formula generali cum pro variis valoribus ipsius m, 
turn indicibus et superioribus et inferioribus omnimodis permu- 
tatis, permultae aliae similes formulae profluunt.” 
The only other point to be noted at present is contained in the 
casual remark that the /3’s may be expressed as differential coefficients 
of A. When dealing later (p. 20), with a special form of determinant, 
he says — 
“ Data occasione observo generaliter, si a K ^ et a\ tli inter se 
diversi sunt, propositis n aequationibus linearibus hujusmodi : 
a \,l U l + a l,2^2 + ••••+ a l,n U n = V 1 j 
a 2,l^l "t" a 2,1^2 + •«••+ a 2 ~ ^2 > 
a n,l U l + a n,2 U 2 +••••+ <*> n,n U n = V ni 
statute 
T= a 14 a 2>2 .... a n>n , 
sequi vice versa : 
