488 
MR. R. LACHLAN ON SYSTEMS OF CIRCLES AND SPHERES. 
the result of substituting the coordinates of the pole of L in M, then the anharmonic 
angles 9, <f> of the two conics are defined by :— 
l-R= v /(l-S , )(l-S ,, ).cos 9, 
1 +R= v 7 (1 —S')(l —S").cos </>. 
A proof of a theorem similar to that given in § 6 of the present memoir is to be 
found in Salmon’s ‘ Conics,’ p. 366. 
10. Let us take as the equations of the three conics 
S=cr+?/ 3 H-2 3 =0, 
a~ (ad+r+ z2 ) — (xf + yg J r zhf= 0 , 
a'-(ot?+y Z ' -f z s ) - (xf+yf+zh'f= 0. 
Then following the method used in Casey’s paper (§ 126), we form the discriminant 
of 
aS h ± (xf+ yg+zh ) + X {AS* ± (xf+yg '+ zh ')} = 0 ; 
and we obtain 
(a'~ —S')X 3 + 2 (aa — Tt)A -f a 3 —S = 0 ; 
and so, if we take 
act — It = \/(A 2 —S')(a 3 —S).cos 9, 
where ~R=ff-\-gg’-\-hli , S=/ 2 +r/ 2 +//, 2 , S'=f' 2 -\-g' 2 -\-h'' 2 ; we have for the tact- 
invariant of the conics crS —L 3 , a' 3 S — M 3 , (a 3 — S)(a' 3 —S') sin 3 9= 0 ; or 9=0. 
11. Again, forming the discriminant of 
{xf+ yg + zh) +X {a'S T (xf + yg' + zh' } = 0, 
we obtain for the tact-invariant, (f)= 0, where 
aa'+R = \/(a' 3 —S')(a 3 —S) cos </>. 
12. Either of these expressions, ad ± E, might be defined as the power of the two 
7j- 
conics a 2 S—L 3 , a' 3 S—M 3 . For if 9= )5 it is evident, as Casey has shown, that the 
pencil formed by the lines L, M, and the chords of contact of the two line pairs which can 
be drawn to touch S from the points of intersection of a' 3 L — a 3 M with « 2 S — L 3 , is 
77* 
harmonic; and so 9=~ is the condition corresponding to the case of two circles cutting 
orthogonally. In this case the power vanishes. 
