578 
Proceedings of the Royal Society 
sibly account for no advance being made to like identities involving 
four sets of four letters x, y, z, w ; x\ y\ z\ w'; &c. 
In this first memoir the algebraical identities are brought together 
and stated at the outset as follows : — 
“ Lemme. 
“ 1. Soient neuf quantity quelconques 
x , y> % x \ y\ x ", y", «" 
je dis qu’on aura cette equation identique 
(xy'z" + yz'x" + zx'y" - xz'y" - yx'z" - zy'x " 2 ) 
= ( x 2 + y 2 + z 2 )(x' 2 + y" 2 4- z' 2 )(x" 2 4- y' ' 2 4- z") 2 
4- 2{xx + yy + zz')(xx" + yy" + zz")(x f x" + y'y" 4- z'z") 
- ( x 2 + y 2 + z 2 )(x'x" 4- y'y" + z'z") 2 
- (x' 2 4- y' 2 + z' 2 ) (xx" + yy" + zz") 2 
- ( x " 2 + y" 2 + z" 2 )(xx' + yy' 4- zz!) 2 . 
“ Corollaire 1. 
“ 2. Done si Ton a entre les neuf quantity pr^cedentes ces 
six equations 
x 2 +y 2 +z 2 = a x'x" + y'y" + z'z" = b, 
x' 2 4- y' 2 +z 2 =a' xx" 4- yy" + zz" = b f , 
x" 2 + y' 2 + z" 2 = a" xx! -\-yy' +zz = b", 
et qu’on fasse pour abr4ger 
£ = y'z" - z'y " , y = z ' x " - x ' z " > £ = x 'y" ~ y' x " ? 
/ 3 = J(aa'a" + 2 bb'b" - ab 2 - a'b' 2 - a"b " 2 ) ; 
on aura 
x£+yrj + zi = /3. 
On aura de plus les Equations identiques suivantes 
x '£ + y'y + = d , x"i + y"rj + z"£ = 0 
£ 2 + y 2 +F = a'a"-b 2 , 
y’l - z'r i — bx' - a'x" , y"l - z"rj = a"x' - bx" , 
z'£ - x'Z, = by' - a'y" , z"£ - x"t = a"y' - by" , 
x'y - y'£ = bz' - a'z " , x"rj - y"£ = a"z' - bz" , 
qui sont tres faciles k verifier par le calcul. 
