4 
PEOFESSOE K. PEAESON OX THE INFLUENCE OF NATLTEAL 
Let us consider the quadric of the order 
Q = Sj {CpjjXj^) + 28^ {cpfjXpX^ = constant, 
and fix our attention on two of the variables, say the first two, and ifo. If these 
Ije considered constants, the quadric of the remaining n — 2 variables will not now 
be referred to its “ centre.” But its centre, x^ . . . x,„ will be given by the 
equations ; 
CisZi + CojXo + ^330:3 + C43X4 + . . . + = 0. 
CiU’i + C24.'r2 + C34T3 + 044X4 fi- . . . + c,nX,i — 0. 
Cln^l "h C2;,X2 fi- 03^X3 fi- C4,,X4 + . . . + - 0. 
The following expressions will not be zero, but will be written a and /3 
(ii.). 
CiiXi + C21X2 4- 031X3 + 04i.'r4 + . 
O 12 X 1 + O 22 X 2 -{- O 32 X 3 4- O 42 X 4 4- . 
Now, if A be the discriminant. 
4" CniX,i, — a. 
4~ CiioX)^ — /3. 
(in-)- 
^12, 
'135 
^215 <^225 ^235 
<^315 ^*325 ^35 
Cm 
Cm 
Czn 
C/ll) C/,2, ^7535 • • • C/, 
and Cpg the minor corresponding to Cp,j, we have by solving the above 7i linear 
equations. 
_ H /SC 12 
^ ’ 
whence : 
Generallv also : 
a — 
A (l^iCoo ^2^ I 2 I 
n ri _ 01 3 
'“n'^23 ^13 
^C]o "t /50.70 
'a 
O _ A 11 X 1 C 12 ) 
^ “ C 11 C 22 - Ci2“ 
(v.). 
Qi f, 
C’lo + /3C 
2? 
(C 1 A 3 - CoyLi alU + ( Cn A - C i,Ci3)^’3 
C ~ 
'"12 
ri c 
L. 22 
(vi.). 
—. iliyXi 4 - D 2 JX 2 , say. 
