580 
Proceedings of the Boyal Society 
are essentially the same ; but no one can deny that the latter con- 
tains on the face of it an all-important fact which is hid in the 
former, and which in Lagrange’s time could be made known only by 
an additional statement in words. 
The second identity 
x'£+y'r) + z'£= 0 
is a simple case of one of Vandermonde’s, viz., that regarding the 
vanishing of his functions w r hen two of the letters involved were the 
same. 
The third identity 
£ 2 + ^ + £ 2 = a! a" - b 2 
is in modern notation 
y' y" 
2 
+ 
z' z" 
2 
+ 
x' x" 
2 
x' 2 + y ' 2 
+ z' 2 
x'x" + y'y" + z'z" 
z' z" 
x' x" 
y' y" 
x'x" + y'y" 
+ z z 
x" 2 + y" 2 + z" 2 
and is thus seen to be a simple special instance of a very important 
theorem afterwards discovered. 
The fourth identity 
y't, - z'rj = bx — ax ", 
may be expressed in modern notation as follows : — 
1 y ' 
2 ' 1 - 
x'x" + y'y" + z'z" x" 
Ik*" i 
I^V'll 
x' 2 + y' 2 + z' 2 x' 
and, quite probably, has also ere this been generalised in the like 
notation. 
The fifth identity 
+ (a"b" - bb')x' + {a'b’ - bb")x" 
x , 
a 
is not so readily transformable, the determinantal theorem which it 
involves being indeed completely buried. Multiplying both sides 
by a ; then doing away with a, which seems perversely introduced 
“ pour abr4ger” when no like symbol of abridgment takes the place 
of a"b" - bb' or of a'b' - bb" ; and transposing, we have 
P£ = x(a'a" - b 2 ) - x\a"b" - bb') + x"(bb" - a'b') 
x' a' b 
x" b a" ; 
