534 
MR. R. LACHLAN ON SYSTEMS OP CIRCLES AND SPHERES. 
Hence 
_r__cjr 
b — a a? a 
(U5) 
98. The double tangents which belong to this system of bitangent circles are 
given by 
fj? cv 2 4 vp 
b — a a? a 
= 0, 
and 
_2p_ 0 
r, e e 
If <j> be the angle between them, we shall have, since 
—+— — — 
„„q l o O 3 
r<r 
, L-/A + ± + « 
an 1_5—a\n 2 ‘ r 0 3 ‘ e 2 
tail 0= 2 c 4 ' 4 ! 
9 9 9 9 I 7 9 
«?v ere - 1 ae" o —a 
(116) 
Hence these double tangents cannot coincide unless 
a , b , c 
: + n> + 7^ = ( h 
?V 
in which case, the double tangents from the centre of the other principal circle 
coincide also; but this equation is the condition that the curve should be a cubic. 
99. If in (112) we take k=0, we have a series of tangent circles passing through 
the node, two of these circles will reduce to straight lines, which will be the tangents 
from the node; in this case we shall have 
whence 
where 
A fji v —2 p 
ax' by' 0 cz' ’ 
q O 1 q 
r + /i + te = 0; 
a b c 
A_i_y—?£=0 ; 
? 2 6 
and the angle between them is given by 
tan (f)= 
-4e/ a b c 
2+.72+H 
abc \?’j 2 r 2 3 e 2 
- + - + 1 
c^a^b 
These also coincide if the curve is a cubic. 
