51G MR. R. LACHLAN ON SYSTEMS OF CIRCLES AND SPHERES. 
Whence, if the two circles cut at an angle </>, we have 
where 
cos (f) = 
a 
a'p 7/ a^ 
a« +& db 
,3^ 7/ 
-+■ c - 0 ^- + d 
ap 
del 
2 v/N'K b, c, d) .¥(»', V, d, d!) ’ 
'i'(o, b, c, d)= — 
a i,n 
tt l,2’ 
a i, 35 
«1,4» 
a 
a>2 ]_> 
a 2, 2> 
^2,35 
^2,45 
b 
a 3,1’ 
a 3, 2> 
Cl 3, 3’ 
a 3,45 
c 
a 4,n 
®4,2) 
a 4,35 
a 4,45 
d 
a, 
b. 
c, 
0 
(81) 
The straight Line .—§§ 56-58. 
56. Proceeding as in § 51, we see that the equation to the straight line, wdiose 
coordinates are (X, g, v, p), is 
^ x+ ^y + t z+ % w=0 - .< 83 ) 
But by equation (72) 
hence the equation 
k *-° ; 
ax + hy + cz + dw = 0 , 
will represent a straight line, provided that 
aky — (~ hko d~ eJc% —j - dky — 0 \ 
• (83) 
and if this condition be satisfied the coordinates of the lines are given by, 
where 
d\fr 
d-yjr 
a l a t|t 
dx 
_3/a _ 
0Z5 _ 0p 
= 2 
, /-K.A 
a 
l 
V if 
v' 'F 
A = 
15 
°h,2s 
«1,3. 
«1,4 
J 
^'2, 1 ’ 
oq, oj 
®2, 3j 
«2.4 
a 3,l5 
%, 25 
CO 
CO 
«3,4 
«'4,15 
«4,25 
^4,3’ 
«4,4 
(84) 
and 'F denotes the same expression as in § 55. 
