918 
PROFESSOR A. CAYLEY ON THE SINGLE 
'& ’= — agh, 
33 3 = bhf, 
(&' : = cfg, 
53 = — abc, 
no?*= ®,w, 
(£Hb= — 3353g, 
SU3c = -CTDh, 
Ebcf = -33€B, 
33cag = 
<£fabh = 1&33I3, 
Ofgh = -&U(L\ 
=abcfgli, 
c-B-+b : r : -f : D- = bcf(af+bg+ch), =0, 
—c 3 gt 3 . +a 3 (!£ 3 -—g 3 I3 3 —cag( „ ), =0, 
—b'^°+a' : 33' . — li~33'=abh( ,, ), =0, 
-f°E'+g-B : +ii-e : . =f g h( „ ), =o. 
It is to be remarked that taking c, a, b, cl in the order of decreasing magnitude we 
have —a, b, c, f, g, h all positive; hence £t 3 . 33 3 QP, I3 3 all real; and taking as we 
may do, J3 negative, then £t, 33, (?' may be taken positive ; that is we have—a, b, c, 
f, g, h, ill, 33, (&, — O all of them positive. 
40. We have 
A 3 0 = -j- /3~ = f, 
B 3 0= 2a/3 = 33g, 
D 3 0 — a~—/3~= CTh. 
The foregoing equations 
^_B 2 0 _ C 2 0 
t — A 2 0 ’ ' — A 2 t)’ 
give 
and we thence have 
k= 
3Sg 
satisfying kr -f- k'~ = 1. 
41. Observe further that substituting for 
£t 2 =c —5.6 —cl.c —cl, 
33 3 =c — a. c — cl. a — d, 
GH 2 =a — b.a—d.b —d, 
13 : — c — b.c — a.a —b, 
a, b, c, f, g, h their values, we have 
= c—d.d—b.b—c, 
= d—a.ci—c.c—d, 
— — .a—b .b — d.d — a, 
= —.b — c .c — a.a — b. 
where in the first set of values all the 
differences are positive, but in the second 
