Fraser — Reduction of a Quartic Surface to a Canonical Form. 83 

 iii. Shows that the circle cuts the fixed circle 



x' + y -¥ 



2^ 



X + 



2/ 



orthogonally ; but, since 



a + \ ^ + A 



r 





« + Ai " J + Ai 4 ' 



r 



« + Ao 



0, 



2(712 



2/2 



_ Ai + Aa _ 



{a H- \,){a + Ao) ' {b + Ai)(6 + A^) 2 ' 



hence the fixed circles are orthogonal, and as in the case of the binodal 

 quartic the 16 points of intersection of the tangents from the circular 

 points to the quartic lie by fours on these circles, 



2! 



c. 



'f'{K) 



where 

 and 



2(7 2/y A,n- „ 



{^' + y-) + — -,- X + ~~ + --f =0, 



r, = (« + A,)(J + A,) ; 



^4 ^ (« + A^)(5 + A^) 



f/ , OS 2ff« 2/y A,^,^ 



^ -^ ^ « + A^ J + A^ 2j ' 



{X^ + y2)2 _|, ^^2 ^ Jy2 ^ ^Igy, ^ 2/// + C, 



ere /(A,) = 



« + A^ ^ 





ff ^ + A^ / 







The reduction is just the same 



C = (« + A)(J + A), 



-i^=^(i + A), 



-(?=/(« + A), 





&c. . . . 



= 0. 



