ME. A. CAYLEY’S SIXTH MEMOIE UPON QUANTICS. 
71 
is of course 
177. The equation of the line passing through the points (a, b, c), {a', h\ d), is 
X , 
a , 
a', 
b, 
b\ 
and if in this equation («', b\ d) are considered as indeterminate, we have the equation 
of a liae subjected to the single condition of passing through the point («, 5, c). The 
equation contains apparently two arbitrary parameters, but these in fact reduce them- 
selves to a single one. 
178. The coordinates of the point of intersection of the Lines 
oiX + 7 ^ = 0 , 
are given by the equations 
a'^-f/3'y+7'^=0. 
a*, y, 2=187' — (3' 7, yoi'—y'u, a(3' — a'^; 
and if in these equations we consider a', f3', 7' as indeterminate, we have the coordinates 
of a point subjected to the single condition of lying in the line ocx~i-/3^-^yz=0 ; the 
result, as in the last case, contains in appearance two arbitrary parameters, but these 
really reduce themselves to a single one. 
179. The condition in order that the points (a, b, c), (a', b\ d), (a", b”, d') may lie in 
a line is 
a , b , 
a', b', 
a", b", 
which may also be expressed by the equations 
d ^ b \ c" — yd) ^ \c-\-yijd^ 
where X, /a are arbitrary multipliers ; these equations give therefore the coordinates of 
an indeterminate point in the line joining the points (a, b, c) and («', b\ d). 
180. The condition that the lines 
ux +72 =0, 
k'x 4-|S'y +7^2: = 0, 
oi"x-\-^''y-\-y''z=(i 
^ 5 y 
/3', 7' 
/3", 7" 
h 2 
may meet in a point is 
