356 Mr. Rutherford’s Demonstration of some useful 
(PQ Ra ~ +5451 Fad 
1 
L 
(PQR) 4... Saeed rae 
ig 
Now if we take the swm of the equations (1.) and (5.), (2.) and 
(6.), (3.) and (7.), (4.) and (8.), we shall, in each case, have the 
same resulting equation, viz. 
U i ul 1 u 1 
Sa ett e+ghyt4oa ba =, (9x) 
= Ai j BB ; At aa 
Let ST be the line of intersection of the two planes in (1.) 
and (5.); that is of the planes PQ R and P’Q'R’; then the 
line S T being common to the planes P Q R and P’Q'R’, and 
since the point I is common to the planes P'Q R and PQ'R’, 
the point H to the planes P Q’R and P’QR’, and the point 
G to the planes PQ R’ and P’Q’R; therefore it is obvious 
that equation (9.), which is the sum of the equations of these 
planes, taken two and two, is the equation of the plane which 
passes through the line S T and through the three points 
G, H, I the points of intersection of the diagonals of the 
quadrilaterals formed by the two planes PQ R and P'Q'R! 
with the axes of coordinates. 
Cor.—If the equations of two planes be denoted by 
EME Bo 7 endo ah eed 
oa B fi ao BN 
then will the difference of these equations, viz. 
(2-2 }e+{5—q}yt {p—-2hene 
be the equation of the plane which passes through the line of 
intersection of these planes, and the origin of coordinates. 
The principles developed in these theorems and corollaries 
are very effective in analytical inquiries, and in order to point 
out their application I shall add the following demonstration 
of a well-known theorem. 
Turorem.—If straight lines A Q, BR, CS be drawn from 
the angles A, B, C of a triangle through any point P to meet 
the opposite sides in Q, R, 8; and if 
QR, QS, RBS be drawn to meet the 
sides of the triangle in G, H, I; then 
will the points G, H, I be in the same 
straight line. 
Take H C, HQ for the axes of x 
and y respectively, and put H A = a,, 
HR=a, HC =e, HS=6,, and HQ 
= B,; then we have the equations of 
the several lines as below. 
