Theorems in the Geometry of Coordinates. 357 
(AB)... +3 =1...(1) (RS)... + Zale) 
(GQ)... 5 +751...) (AQ) ve +g = Lee (6) 
(BC)... +2 =1...(8)|(CS)... + 2=1..- 6) 
a; Bo a; By 
Now (Theorem I., Cor.) if we subtract (4.) and (1.) from (3.) 
and (2.) respectively, we shall have the equations of H I and 
HG, viz.— 
oy aaa tater fa—ghynoee 
Senet 
(HG)... eed em Sa 5 dada 
These equations will be identical if it can be shown that the 
coefficients of « are equal, and to effect this, the condition that 
the lines AQ, BR, CS all pass through the same point P 
must be employed. Hence if we add the equations of CQ 
and A S, viz. (3.) and (1.), we shall have the equation of B P 
(Theorem I.), viz. 
ars er {etaly=2. sue 9) 
To find where this line cuts the axis of z, make y = 0, and 
we get 
=> T 
a) a3 
and this must evidently be the value of H R, because BPR 
is a straight line by hypothesis; but H R = 2, and therefore 
we must have 
=| 5 OL ae - 
= ones as e ay as 
a) 2s 
1 1 1 1 2 
. — -—=— ——, by transposition ; 
Cg oe ij Fao Se 
and consequently equations (7.) and (8.) are identical, and 
therefore the points G, H, I range in the same straight line. 
The form of the equation of a straight line used in these 
inquiries is well adapted to the demonstration of a class of 
theorems concerning the intersections of straight lines, of 
which the preceding example is an instance, and a variety of 
others will be found in the forthcoming second volume of 
Hutton’s Course, by my talented colleague Mr. Davies, 
