256 
MISS C. A. SCOTT ON PLANE CCBICS. 
and as KK' are harmonic with regard to OH, X in the equation 
(KGHOK'} = [I 3 I 1 H 3 H 1 X} 
must be such that I 3 X may be harmonic with regard to H^Hg; he., X must be H,, 
therefore 
{KGHOK'} = {I 3 I 1 H 3 H 1 H 2 } ; 
therefore 
{KGOK'j = {I 3 I 1 H 1 H 3 }. 
Now the foci of the involution OG, KK', are k, k ; call the foci of I^H^ IjHo, I3H3, 
X, X ; from the relation just proved we have 
{GOKK'A-/c} = {IiHiLH^^’a:'}. 
We wish to j)i'ove {OKA’/c} equianharmonic; i.e., we have to prove {H^I^auf} 
equianharmonic, for which it suffices to show 
{IgHjCca;'] = {Igaj'H^a:}. 
Consider the IH involution, whose foci are xx. From the way it is constructed 
(viz., three points I, their harmonic conjugates H), we know that any cross-ratio in 
the I’s and H s is unaltered 
( 1 ) by any interchange of the suffixes, 
( 2 ) by the interchange of I and H. 
It is convenient to write 1,1', for I^, H^ &c. 
We have to prove 
[2l'xx\ = {2xl'x]. 
We know that xx, 12 , 1 ' 2 ' are harmonic with regard to 33', and therefore in 
involution ; therefore 
{ 121 'a;} = { 212 'a:'} = { 2 'x' 2 l}.(i). 
Now [ll'xx] is harmonic, as also {2'213} ; applying these to (i) we have 
[Ul'xx] = {2'x213} .(ii). 
Again, {121'3} is harmonic, as also {2'x'2x] ; applying these to (ii) we have 
[121' XX 3] — [2'x2l3x}, (hi). 
