226 
Proceedings of Royal Society of Edinburgh. [sess. 
The constitution of the 3rd section is quite like that of the 
others, the first paragraph dealing with the multiplication- theorem 
for the case of determinants of the 3rd order, the second paragraph 
with the same theorem for determinants of the 4th order, and the 
remaining eight paragraphs with geometrical applications. The 
mode of proof of the multiplication-theorem is partly indicated by 
saying that any particular case is made dependent on the case imme- 
diately preceding it ; but its exact character can only be understood 
by a somewhat minute examination. The investigation for the case 
of determinants of the 3rd order stands as follows (p. 29) : — 
“Si considerino i due polinomj 
•X'mfyrfip ~~ Vp^n) 4 " X n (Z m y p “ Vm^p) 4 " X' p fy m Z n ~~ V n^m) 
^ = 2/w) %>} J 
®a(yM -y&) + x h {z a y c - z c y a ) + x c (y a z b - y b z a ) 
= R, Vb , ^cl- 
Se ne effettui il prodotto, il quale, mediante l’equazione (a) del 
primo articolo, si potra disporre sotto la forma seguente 
x m x a (y n y b )(y p yf) 
+ x n x a (y m y c )(y p y b ) 
+ x p x a (y m y b )(y n y c ) 
+ x m x b (y n y c )(VpVa) 
(h) + x n x b (y m y a )(y p y c ) 
+ x p x b (y m y c )(y n y a ) 
+ x m x c (jj n yfj(^y p yfj 
+ % (y m y b )(y P ya) 
+ x p x c {y m y a )(y n y b ) 
Esaminando ora la quantita 
x m x a {y n yf){y p yf) 
Va{y m y b )(:ypyc) 
x p xfy m y c ){y n y b ) 
x m x b (y n y a )(y p y c ) 
x n x b (y m y c ){ypya) 
x p x b (y m y a )(y n y c ) 
x m x c (y n y b )(y p y 0 ) 
x n x c (y m y a )(y p y b ) 
x p x c (y m y b )(y n y a ) . 
x m x a {x n x b (y p y c ) + x p x c (y n y b ) + x n x b x p x c 
- x n x c (y p y b ) - x p x b (y n y c ) - x n x p x b x c j 
+ x n x a {x m x c (y p y b ) + x p x b (y m y c ) -i- x m x c x p x b 
- x m x b (y p y c ) - x p x c (y m y b ) - x m x b x p x c ) 
+ x p x a {x m x b (y n y c ) + x n x c (y m y b ) + x m x b x n x c 
- x m xfy n y b ) - x n x b (y m y c ) - x m x c x n x b j, 
