912 Proceedings of Royal Society of Edinburgh. [sess. 
“on obtiendra l’equation identique 
(ax 2 + by 2 + 2 cxy)(ax 2 + by 2 + 2cxy) - [axx + byy + c(xy 4- xy) } 2 
= (ab-c 2 )(xy-xy) 2 , 
qni a ete donnee par Lagrange dans les Memoires de Berlin de 
1773”; 
in the latter case, it being explained that 
A = bc-d 2 , B = ca- e 2 , C = ab- f 2 , 
T) = ef-ad, F —fd - be , F — de-cf, 
and 
X = yz - yz , Y = zx — zx , Z = xy - xy , 
“ on obtiendra l’equation identique 
(ax 2 + by 2 + cz 2 + 2 dyz + 2 ezx + 2 fxy)(ax 2 + by 2 + cz 2 + 2c7yz + 2ezx + 2/xy) 
- { axx + byy + czz + d(yz + yz) + e(zx + zx) + f(x y + xy) } 2 
= AX 2 + BY 2 + CZ 2 + 2 DYZ + 2EZX + 2FXY , 
que l’on pourrait deduire de l’une des formules donnees par 
M. Binet dans le xvi e cahier du Journal de VEcole Poly- 
technique.” 
This latter, for the sake of future reference, it is well to restate 
in the form 
X 
y 
z 
X 
y 
z 
a 
f 
e 
X 
a 
/ 
e 
X 
f 
b 
d 
y 
f 
b 
d 
y 
X 
Y 
Z 
e 
d 
c 
z 
e 
d 
c 
z 
A 
F 
E 
X 
X 
y 
z 
X 
y 
z 
F 
E 
B 
D 
D 
Y 
a 
f 
e 
X 
a 
/ 
e 
X 
C 
Z 
f 
b 
d 
y 
f 
b 
d 
y 
e 
d 
c 
z 
e 
d 
c 
z 
A concluding paragraph is devoted to noting that P xx , P x . 
in the fourth theorem are expressible as halved differential- 
quotients of P, viz., 
P x , x =m p > Py,y=m p i P Z , Z = \B>\P, 
P.,BW>X> y P, P,.,= i DJD.P, = i 
