584 
Proceedings of the Royal Society 
— a simple instance of the theorem which gives the relation, as we 
now say, “between a determinant and its adjugate.” 
In regard to the remaining identities which we have numbered 
(1) , (2), (3), (4), we note that (1) and (3) are not new, although (3) 
is here given almost in the form desiderated above (pp. 580-1); 
(2) involves the fact that A is the same function of x, x, x ", y , y\ 
y ", z, z\ z", as it is of x, y , z, x', y\ z\ x ", y", z" ; and (4) may be 
transformed as follows : — 
xA = a£ + V'£' + V£", 
y * 
y' 
y" f , 
x 2 + y 2 + z 2 
xx' + yy' + zz’ 
xx" + yy" + zz" 
y 
y' 
y" 
z 
z' 
} 
so that it may be considered as another disguised instance of the 
multiplication- theorem, the determinant just reached being equal to 
X 
y 
z 
X 
0 
0 
x' 
y' 
z’ 
X 
y 
1 
0 
x" 
y" 
z" 
z 
0 
1 
LAGRANGE (1773). 
[Recherches d’Arithm^tique. Nouv. Mem. de VAcad. Roy. . . . ( de 
Berlin) Ann. 1773 (pp. 265-312).] 
This is an extensive memoir on the numbers “ qui peuvent etre 
representees par la formule B t 2 + C tu + D u 2 ”. At p. 285 the expres- 
sion 
j yy 2 + 2qyz + rz 2 
is transformed into 
Ps 2 + 2Q sx + Hx 2 
by putting y = M.s + Nr , 
and z — ms + nx , 
and Lagrange says — 
“ . . . . je substitue dans la quantite PR - Q 2 les valeurs 
de P, Q et R, et je trouve en effafant ce qui se detruit 
VB-Q? = (pr-q 2 ) (M^-h T m) 2 ; . . . 
