of Edinburgh, Session 1885-86. 
581 
that is, finally, 
I vrff ! • I y'z’ I = 
x xx' + yy 4 zz 
x' x ' 2 + y' 2 + z 2 
x" x'x" + y'y" + zz 
xx" + yy" + zz" 
x'x" + y'y" + z'z " 
x" 2 +y " 2 +z" 2 , 
which we recognise as an instance of the multiplication-theorem on 
putting 
X 
y z 
1 
x' 
x" 
x' 
y' 2 ' 
X 
0 
y' 
y" 
x" 
n tt 
y z 
0 
z 
z" 
for the left-hand member. 
LAGKAXGE (1773). 
[Solutions analytiques de quelques problkmes sur les pyramides 
triangulaires. Nouv. Mem. de VAcad. Roy {de Berlin) 
Ann. 1773 (pp. 149-176).] 
In this memoir also there is a preparatory algebraical portion, 
the subject being the same as before, and the author’s standpoint 
unchanged. Indeed the two introductions differ only in that the 
second is a rounding off and slight natural development of the first. 
In addition to £, rj, £, we have now £, r[, £', f, £" used as 
abbreviations for zy" — yz ', xz" - zx", ; in addition to a, we 
have a, a", /?, /3 f , /3", standing for aa" - b' 2 , aa' - b" 2 , b'b" - ab, 
bb" -a'b', bb' — a"b" ; and X, Y, Z, X', Y', . . . ., A, A', . . . are in- 
troduced, having the same relation to £, g, £, .... a, a, .. . 
as these latter have to x , y, z, x , y , , a, a, ... . Lagrange 
then proceeds : — 
“ 3. Or en substituant les valeurs de £, &c., en x, x , &c., 
et faisant pour abr4ger 
A = xy'z’ + yzx" + zx'y" — xz'y" - yx'z" - zyx", 
on trouve 
X = Ax, Y = Ay, Z = Az , 
X' = Ax', Y' = Ay', Z' = Az', 
X" = Ax", Y" = Ay", Z" = Az", 
done mettant ces valeurs dans les dernieres Equations ci- 
