Useful Theorems in the Geometry of Coordinates. 353 
means a common word among mathematicians; it was a tech- 
nical term of judicial astrology, and would probably be avoided 
by the geometer: Torporley was an astrologer, as appears 
from the opening of his work. Trias and ternio would sug- 
gest themselves first (¢rinitas being excluded for an obvious 
reason). On this word probably the question will turn: if it 
should be found that no mathematicians of the period use the 
word triplicitas except Torporley and Napier, it will be diffi- 
cult to avoid presuming that the latter must have seen the 
work of the former. 
I remain, Gentlemen, 
Yours faithfully, 
University College, March 13, 1843. A. Dre Monrean. 
LIX. Demonstration of some useful Theorems in the Geometry 
of Coordinates. By Wi.LiaM Ruruerrorn, Esq., RAS. 
Royal Military Academy*. 
(THEOREM I.—If the equations of two straight lines be 
wv y a y 
—+4=1land—+4=1 
a p a, py A 
then will the swm of these equations, viz. 
(ed }es (pepe 
be the equation of the straight line passing through the point 
of intersection of these lines, and the point of intersection of 
the diagonals of the quadrilateral formed by the intersection 
of the given lines with the axes of coordinates. 
Let OX, OY be the axes of coordinates 
having any angle of ordination, and let P, Q, nl 
R, S be any four points in these coordinate WIN 
axes; then if OP =a, OQ=24,OS= 6, ,.fst 
and OR = £,, the equations of the lines S P rs 
and R Q, drawn through the points S, P and ee 
R, Q are respectively o 
ie y v JY 
— + = ] and — +e = hs. « » (1 
a B a, | py (1.) 
Now these lines must either be parallel, or they will meet if 
produced. Let them meet when produced in H, and let I 
be the point of intersection of the diagonals S Q and R P of 
the quadrilateral PQRS. Join OH, and through H and I 
* Communicated by the Author. 
