472 Proceedings of Royal Society of Edinburgh. [july 18, 
a, ft y 
d, ft, y 
a , , 7 
atque / implicare ipsam g, siue sub f contentam esse. Ex 
tali itaque snppositione sponte sequuntur sex equationes pro 
sex coefficientibus in g, quas apponere non erit necessarium : 
hinc antem per calculum facilem sequentes conclusiones 
eitoluuntur : 
“ I. Designato breuitatis caussa numero 
a /3'y" + ft/ a/' + yd — y fi'd' - a y ($" — fidy" 
per k inuenitur post debitas reductiones 
E = MD, ...... (xxii. 2.) 
• •••••••• 
When freed from its connection with ternary quadratic forms the 
theorem in determinants here involved is 
If A 0 = a 0 a 0 2 + a x ai + a 2 a 2 2 + 2& 0 a 1 a 2 + 25 1 a 0 a 2 +26 2 a 0 a 1 , 
Ai = UqPq" + + 2& 0 /3j/ 3 2 2 G^o^ 2 + > 
A 2 = a o7o 2 + a i7i + a - 2 l 2 + 2& o7i7‘2 + 2& i7o7 2 + 2& 27o7i > 
B 0 = a 0 £o7o + + %027 2 + Wi7* + /3 2 7i) + ^i(^o72 + £ 2 7o) + ^(0o7i + #i7o) 
Bj = a 0 a o 7 q + «i«i7i + « 2 «272 + & o( a i72 + « 2 7 i> + & i(«o 7 2 + a 2 7o) + & 2 («o7i + «i7o) 
Bo = ^o a O^O ®l a I$l "t ttoC*o^2 + &o( a l$2 "t °2^l) "t ^l( a 0^2 "t" a 2$o) ^ 2 ( a 0$l "t a I@o) j 
then 
A 0 B 0 2 + A 1 B 1 2 + A 2 B 2 2 - A 0 A 1 A 2 - 2B 0 B 1 B 2 
= (a 0 6 0 - + 4 a.ff — agxyi^ — 26 0 & 1 & 2 )j 
X (a 0 ft y 2 + ft)7l a 2 + 7o a i/^2 ~ 7oA a 2 — a o7ift — /Vite) 2 * 
As thus viewed it is an instance of the multiplication-theorem, the 
product of three determinants (in the modern sense) being ex- 
pressed as a single determinant. 
The multiplication-theorem is also not very distantly connected 
with the following other statement of Gauss : — 
“ Si forma ternaria / formam ternarium f implicat atque haec 
formam f": implicabit etiani / ipsam /". Facillime enim per- 
spicietur, si transeat 
/ in /' per substitutionem 
A 7 
a, 
f in f" per substitutionem 
e, t 
