102 ON LINEAR TRANSFORMATIONS. 
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where the symbols £#, ye refer to the two systems x 2 , y 2 : x 3 , y 3 . Thus it is easily 
that we may write 
& = & + &, V9 = V*+V3, or 1^=12 + 13 = 12-31, 
whence the function becomes 
(l2-31) 4a 23 * a UVW, 
of which all the terms vanish except 
X2“23 , ‘3i“fTVW. 
[2a]-“ 
Hence putting 
[4a] 2 “ _ 2 4 “ 1.3 ... (4a- 1) 
~[2a] 2 “~ 2.4... 4a ’ 
we have B ia [U, S 2a (F, W)]=KG a (U, V, W), 
or in particular 
B*[U, B*(U, U)] = KG a (U, U, IT). 
Thus for example, neglecting a numerical factor, 
(<ax 2 + 2 bxy + cy 2 ) (cx 2 + 2dxy + ey 2 ) — (bx 2 + 2 cxy + dy 2 ) 2 
= (ac — b 2 ) x* + 2 (ad — be) ot?y + (ae + 2bd — 3c 2 ) x 2 y 2 + 2 (be — cd) xy 3 + (ce — d 2 ) y 4 , 
and then 
e (ac — b 2 ) — 4cZ f (ad — be) + 6c £ (ae + 2bd — 3c 2 ) — 46 £ (be - cd) + a (ce — d 2 ) 
= 3 (ace — ad 2 — b 2 e — c 3 + 2&ccZ). 
We have likewise the singular equation 
d 
R M (V, W) = K^*^y* + y^)c.(U, V, W) 
where 
U = (a 0 ^ a - aa« 4 “- 1 ^ ... + a 4a y 4 “) , &c. 
If however U= V — W, we must write 
d . d 
B.^U, U. U), 
seen 
the reason of which is easily seen. This subject will be resumed in the sequel.