CONTACT AT ANT POINT OF A PLANE CUE'^^. 
375 
H= a, h, g 
h, b, f 
be the Hessian ; if, moreover, 
<I)=— X, |M;, V 
K a, h, g 
gj h, b, f 
. ^ 9^ 
be the bordered Hessian (we may also write 0=(^, B, C, €>, where 
(31, 35, C, JT, are the inverse coefficients of («, b, G,f, g, h), viz. ^=z(bc—f^) See.); 
and finally, if for shortness we write 
n=b,<i> 
□ =B,H .B^O+B.H .B^0+B,H.B,0, 
then we have 
p ^ 
^ m— 1 
<!>, 
0^= 
P3=-^D<i>, 
® m — 1 
(m — 1)^ 
1 
Q3 = 
P4=-^D^o-, "^^h, Q,= 
In the present case U = 0, and we have 
BiU=: — Q3B^ 
BtU=-Q4B^ 
11. Hence, substituting for Q 2 and their values, the first and second of the equa- 
tions for the five-pointic contact give 
p=2(m-2). 
md 
{m — l 
1 
(to— l)^ 
H, 
^DH, 
FH- 
( to — 1 
3 (to — 2) TT.T. 
+ — rv3H€>. 
r (to — 1 )'^ 
B,P= 
.Dh 
3 H ’ 
and observing that H is a linear function of (X, Y, Z), and consequently that P, B,P 
denote simply the values which H assumes when (w, y, z), (dw, dg, dz) are respectively 
substituted for (X, Y, Z), we see at once that these two conditions will be satisfied if we 
put 
n=f^ HH+A.HU, 
where A is an arbitrary constant, or, what is the same thing, an arbitrary function of 
(a’, y, z). We have thus the general equation of a conic of four-pointic contact. 
12. The above value of H gives 
a.P==ts3.H+A3,U, 
H 
3 
^ D 2 
