PROFESSOR CAYLEY ON POLYZOMAL CURVES. 



we have 15 = 25 = 35 = 45 = ; and the equation becomes 



103 







T2 2 , 13 2 , 1? 



2~T 2 , , 23 2 , 24 2 

 3T 2 , 32 2 , , 34 2 

 41 2 , 42 2 , 43 2 , 



which is the relation between the distances of any four points on a cone-sphere ; 

 this equation may be written under the irrational form 



= 0, 



23 . 14 + 31.24 + 12. 34 = 0. 



Taking (a, a\ a"), (b, b', b"), (c, c', c"), (#, y, z) for the co-ordinates of the four points 

 respectively, we have 



23 = s/{b-cf + (b'-c'f + (b"-c"f, 14 = J(x-af + (y-a'f + (z-a"f , 

 31 = J{c-af + {c'-a'f + {c"-a"f , 24 = J{x-bf + {y-b'f + {z-b'J , 

 12 = J {a -If + {a'-b'f + (a" -bj , 34 = J(x-cf + (y-c'f + {z-c"f , 



or the symbols having these significations, we have 



23. 14 + 31. 24 + 12. 34 = 



for the equation of the cone-sphere through the three points ; or rather (since the 

 rational equation is of the order 4 in the co-ordinates {x, y, z)) this is the equation 

 of the pair of cone-spheres through the three given points ; and similarly it is 

 in the first problem the equation of a pair of circles each touching the three 

 given circles respectively. 



In the first problem the radii of the given circles were i(z — a"), i(z — b "), 

 i(z— c") respectively; denoting these radii by a, /3, <y, or taking the equations of the 

 given circles to be 



{x-af + (y-a'f — a? = 0, 

 (x-bf + {y-b'f -jS* = 0, 

 (x-cf + (y-c'f - 7 2 = 0, 



the symbols then are 



23 - J(b-cf +(b'-c'f-((3- 7 f, 14 = J(x-af + (y-a'f - a 2 , 

 31 = J(c-af + (c' -a'f - ( 7 -ecf , 24 = J(x-bf + (y-b'f - /3 2 , 

 12 = J(a-bf + (a'-b'f - (a-/3) 2 , 34 = J(x-cf + (y-c'f - y , 



and the equation of the pair of circles is as before 



23.14 + 31.24 + 12.34 = 0; 



