ME. A. CAYLEY’S SIXTH MEMOIE UPON QUANTICS. 
73 
183. To find the equation of the line passing through the point of intersection of the 
lines 
and the point of intersection of the lines 
Write for shortness Uy—a^x-\-^iy-\-yiZ, &c. ; then we have identically 
= 0 , 
and the two equations 
Ui , 
U 21 
M 3 , 
M 4 
Kj , 
•> 
«4 
13. 
/34 
yn 
72, 
73, 
74 
M], 
M2, 
• ? 
• 
= 0, 
• ^ 
• ? 
M3, 
M4 
«2, 
®^3, 
“4 
Oil , 
0^2, 
®^3, 
^4 
^2, 
1 ^ 3 , 
/34 
^1, 
^2, 
^3, 
/34 
7. 
72, 
73, 
74 
7i, 
72, 
73, 
74 
= 0 
are consequently equivalent to each other, and each of them represents the required line. 
It is easy to deduce the form 
= 0. 
X 
«2, 
• 5 
• 
• D 
+2 
7i, 
72, 
• 9 
• 9 
®^3, 
“4 
®i. 
®^2, 
^39 
“4 
a„ 
®2, 
“a. 
^4 
^2, 
^3, 
• , 
• , 
fSz, 
/34 
7i, 
72, 
73, 
74 
7i, 
72, 
73, 
74 
• 5 
• 9 
73, 
74 
184. The condition in order that the points of intersection of the lines Mj = 0, 
of the lines % 3 = 0 , ^ 4 = 0 , and of the lines ^ 5 = 0 , W6=0 (where, as before, denotes 
aia;+|3,2/d-7i2, &c.) may lie in the same line, is 
= 0 , 
«!, 
®^2, 
^3, 
054, 
• 9 
• 
^2, 
/34, 
• 9 
• 
7i, 
72, 
73, 
74, 
• 9 
• 
* 9 
• 9 
*3, 
a4. 
“5, 
“6 
• 9 
9 
/35, 
(^6 
• 9 
• 9 
73, 
74, 
75, 
76 
which is of course really symmetrical with respect to the six sets. The last formula 
was given by me, ‘ Cambridge Mathematical Journal,’ t. iv. p. 18 (1849). 
185. Instead of the term point of a curve, it will be convenient to use the term 
‘ ineunt’ of the curve. 
The line through two consecutive ineunts of the curve is the tangent at the ineunt. 
The point of intersection of two consecutive tangents is the ineunt on the tangent. 
