354 Mr, Rutherford’s Demonstration of some useful 
draw the straight line HI. ‘Then the equations of the dia- 
gonals P R and S Q are respectively 
24 2=1ad—4+%<1; . 
at Rtg F Mega uti 8d 
and since the point H is common to both the lines S P and 
RQ; and the point I common to both the diagonals SQ 
and PR; therefore the swm of the equations in (1.), viz. 
{Eelber{ped}an. 
is evidently the equation of a line passing through H the 
point of intersection of the lines denoted by the equations (1.); 
but equation (3.) is also the sum of the equations (2.), and 
therefore equation (3.) is likewise the equation of a line pass- 
ing through I the point of intersection of the diagonals of the 
quadrilateral PQES; hence the truth of the theorem is 
established. 
If the lines SP and RQ are parallel, then the triangles 
OPS, OQR will be equiangular, and therefore we have the 
relation 
a 6p hy 
SS —— or =— —! . 
ay By B, a Bs 
and this value being substituted for 6, in equation (3.), gives 
or Ua cays Mod We Weed Vy he a 
oe a + gh alo = Aer a ag re Fae aaa 
which is the equation of the straight line parallel to the given 
line 
Ei aay igo 
toh 5 B _ 1, 
and passing through I the point of intersection of the dia- 
gonals of the trapezoid PQRS. 
Cor.—Hence it is obvious that the difference of the equa- 
tions in (1.), viz. 
{Pot}es(p-p}emo 2 
is the equation of the straight line O H passing through the 
origin of coordinates, and the point of intersection of the two 
lines S P and RQ. If these lines are parallel, then we have 
B, = B; 
and by substituting this value of 8, in equation (5.), we have 
a—-— arr re ae 1 seh 
" {tgp Hore te 6 PO Re a | 
