or A CTJEVE OF THE FOTJETH OEDER. 
359 
As already noticed, it has been shown by Hesse (and his demonstration is to be here 
reproduced) that the two forms ni = 0 and n2=0 are equivalent to each other. And 
the object of the memoir is to show that the third form n 3=0 is equivalent to the other 
two. The equivalences in question subsist in virtue of the equation U=0, that is, the 
functions Hj, Hg, Ha are not identical, but differ from each other by multiples of U. 
Demonstration of Hesse’s Theorem. 
Let {a, b, c,f, g, h), {a!, b', g\ h') be any systems of coefficients of a ternary 
quadratic function ; (a, b, c, f, g, h), (a', b', c', f', g', h') the reciprocal systems as above, 
[x, y, z) arbitrary quantities. Consider the function 
□ =(«, b, c,f, g, hXoo, y, z)\ (V, b', c', f', g', h'X«, b, c, 2/, 2g, 2h) 
— (a', b', c', f', g', B ! Xax -\- hy -\- gz , hx+by+fz, gx-]rfy+cz)\ 
The term involving A' is 
a{a, b, c,f g, hjx, y, zf—{ax-]rhy+gz)\ 
which is 
={ab- h^)f + {ac—g^)z^ + 2{af—gh)yz 
=Cy^-{-^z^-2'Fyz-, 
and the term invohing 2F' is 
/(«, 5, c,/, g, hJx, y, zf—{hx-\-by-\-fz){gx-\-fy+cz), 
which is 
={af-gh)xf+{f - bc)yz-^{fg—ch)zx-\-{hf—bg)xy 
= — Far" — Kyz + Hzor + Gxy ; 
and the entire expression for □ is thus 
A!{Cf-\-Bz^-2'Fyz) 
+ ^\Az^-\-Cxf—2Gzx) 
+ C'(Br^+A/-2Hary) 
+ 2F' (— For^— Ay^+H^or+Gor^) 
+2G'(— Gj/*— +Fay +Hy2) 
-Gxy -\-Gyz-\-Yzx ) ; 
or what is the same thing, 
□ =(BC'+B'C-2FF, CA'+C'A-2GG', AB'+A'B-.2HH', 
GH'+G'H-AF'-A'F, HF+H'F-BG-B'G, FG'+r'G-CH'-C'HJ^, y, z)\ 
which is really the fundamental theorem. It is however used as follows ; viz. the right- 
hand side being symmetrical in regard to the two systems 
(«, b, c,f g, A), {a!, b\ c',f, g’, h'), 
the left-hand side, which is not in form symmetrical as regards the two systems, must be so 
in reality ; or if □' is what □ becomes by interchanging the two systems, then ; 
MDCCCLXI. 3 D 
