ME. E. LACHLAN ON SYSTEMS OF CIECLES AND SPHEEES. 
523 
Bitangent Circles .—§§ 73-75. 
73. A circle 
x 
§ y+y d„' +z dz' +w ho'r +l '^- 0 ’ 
will clearly touch the curve 0 at the point x", y", z", w" if 
00 , 00 00 , 00 00 0-0 00 00 
0a/ 0a/_0?/' ' 0?/'_0Y 0z' _ dw ' 0«/ , 
00 , 00 00 , 00 00 , 00 00 , 00 
030 ' 0*" 02/" " 02/" 00 ' 00 0M0 ” 020 
7.e,, if & satisfy the equation 
H(0+£0) = O, ..(99) 
where H (u) denotes the Hessian of u. 
Since this equation is of the fourtli degree in Jc, we infer that there are in general 
four systems of bitangent circles, each circle belonging to any system cutting a certain 
fixed circle orthogonally, the coordinates of this circle being proportional to the minors 
of the constituents of any row of the determinant H(0+/i;0). 
74. If the coordinates of a bitangent circle satisfy the condition which must be 
satisfied by coordinates of any straight line, the corresponding equation will represent 
the double tangents from the centre of the corresponding circle. In general, then, 
there are eight double tangents. 
75. It is clear that, if by any linear transformation of coordinates the equations 
0=0, 0=0 become respectively <4>=0, 0=0, then the same value of Jc must satisfy 
both 
H(0-f/j0) = O and H(cP + /^) = 0. 
Hence the coefficients of the powers of Jc in equation (99) are invariants. 
Equation to Normal at any Point .—§§ 76-79. 
76. Let (f^0j) be the coordinates of any circle which cuts the curve (f)(xyzw) = 0, 
orthogonally at the point ( x'y'zw ), then by equation (75) we must have 
.( l0 °) 
for all values of Jc. 
3x2 
