483 
364] ON A THEOREM RELATING TO FIVE POINTS IN A PLANE, 
for line OB' we have 
1 2/3 (/3 + 1) 1 
x ' У '■ 2 м + 2/3y {M + 2a/3){M + 2/3y) : M+2a/3’ 
for line OC we have 
1 1 27 (7 + I) 
y ' M + 2/3y M + 27a {M + 2/3y) (M + 27a) 
The equations of the lines Oa, 0/3, O7 are of course (y = 0, z = 0), (z = 0, x = 0),. 
(x = 0, y = 0) respectively; and we therefore see at once that the planes {OA', Oa),. 
{OB', 0/3), {ОС, O7) meet in a line, viz. in the line which has for its equations 
111 
X - y : z ~ M+ 2/З7 : M + 2 7 a : M + 2a/3 ’ 
The lines OB', OC, Oa will lie in a plane, if only 
4/3 7 Q8 + 1) (7-+1). 
{M + 2/8 7 ) 2 
that is 
{M + 2/З7) 2 = 4/З7 (/3 + 1) (7 + 1), 
or, as this may be written, 
M 2 + 4/3y (a + /3 + y + l+ /З7) = 4f/3y {/3y + /3 + 7 + 1); 
that is 
M- + 40Г/З7 = 0, 
or, what is the same thing, 
(a + /3 + 7 + l) 2 + 4a/37 = 0; 
and from the symmetry of this equation we see that, when it is satisfied, 
the lines OB', OC, Oa will lie in a plane, 
„ OC, OA', 0/3 
„ OA', OB', O7 „ „ ; 
viz. this will be the case when the point 0 is situate in the cubic surface 
represented by the last-mentioned equation; this completes the demonstration of the 
solid theorems. 
It is clear that considering five points 1, 2, 3, 4, 5 in a plane, then, since any 
one of these may be taken for the point 0' of the foregoing theorem, the theorem 
exhibited in the first instance as a theorem relating to a triangle and two points, 
and afterwards as a theorem relating to a quadrangle and a point, is really a theorem 
relating to five points in a plane. There are, of course, five different systems of 
points (P, Q, R, S), corresponding to the different combinations of four out of the five 
points. 
Cambridge, March 6, 1865. 
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