58 
ANALYTICAL RESEARCHES CONNECTED WITH 
[114 
| 5. Contains a separate discussion of a system of equations, including as a 
particular case the equations obtained in the preceding section. 
§§ 6 and 7. Contain the application of the formulae for the general system to the 
equations in § 4, and the development and completion of the solution. 
§ 8. Is an extension of some preceding formulae to quadratic functions of any 
number of variables. 
1. Lemma relating to the sections of a surface of the second order. 
If 
anf + by 2 + cz 2 + dw 2 + 2fyz + 2gzx 4- 2 hxy + 2 Ixw + 2myw + 2 nzw = 0 
be the equation of a surface of the second order, and 
+ 33y 2 + + 3Bw 2 + 2 fftyz + 2 (&zx 4- 2 ty^xy 4- 2 %xw + 2 + 2 $.zw = 0 
the reciprocal equation, the condition that the two sections 
\x + py + vz + pw — 0, 
\'x 4- g y 4- v z + p'w = 0, 
may touch, is 
(i^tX 2 + 33/a 2 4- (B>v 2 + î3p 2 + 2 $ pv 4- 2($iv\ + 2|ËJX/a 4- 2 3tXp + 2f\\$Lpp 4- 2 [$tvpf 
x (i5tV 2 + 33/a' 2 + Q&v' 2 + IBp' 2 4- 2$p'v' + 2(£iv'\' 4- 2^X'p 4- 2'%L\'p'+ 2f$[p p + 2ffl,v'p')* 
= (^tXX' + 33/a/</ + QLvv' + IBpp' + Jp {pv' + pv) 4- dir {v\' 4- v'X) + ^ {Xp' + X'p) 
+ % (Xp + x'p) + JÏI (pp + pp) + jE (vp + vp)); 
and in particular if the equation of the surface be 
ax 2 + by 2 + cz 2 + 2fyz 4- 2 gzx + 2 hxy 4- pw 2 = 0, 
the condition of contact is 
j^gtX 2 + 33/a 2 4- (§v 2 + 2$ pv 4- 2dEri'\ + 2|^X/a+ ^ p 2 j 
x (&X' 2 + 33/a /2 + €v' 2 + 2jf pv 4- 2(§v'\' + 2^ 
Jp {pv' 4- pv) 4- dEr (vX' + v'X) + (X/A + X /a) H pp j , 
K ; 
- pp 
in which last formula 
&=bc -f 2 , 3$ = ca-g 2 , <&=ab-h 2 , 
§ = gh-af , d5f = hf — bg, i&=fg-ch, 
K ---- abc - af 2 — bg 2 — ch 2 4- 2/i//i.