209] PLANE REPRESENTATION OF THE THIRD ORDER. 209 



By changing the sign of a, and then changing the signs of ft aud of 7, we 

 shall obtain the four points in which the pencil 0(P 1 , P 2 , P 3 , P 4 ) cuts the 



axis &amp;lt;j&amp;gt; 3 = 0. If four values of j- 1 be represented by k, I, m, n, the required 



02 



anharmomc ratio is, of course, 



(n l)(m k) 

 (n-m)(l-ky 



and after a few reductions we find that this becomes 



But we have 



_ 



Eliminating 0^, we find that 2 2 and 0/ disappear also when we make 



tf = p-2-p s , P a = p,-p 1) y 2 = pi-p,, 

 and we obtain the following result : 



which gives the following theorem : 



Measure off distances p lf p.,, p s , p, from an arbitrary point on a straight 

 line, then the anharmonic ratio of the four points thus obtained is equal to the 

 anharmonic ratio subtended by any point of the p-pitch conic at the four points 

 of indeterminate pitch. 



It is possible without any sacrifice of generality to make the zero-pitch 

 conic a circle. For take three angles A, B, G whose sum is 180 and such 

 that the equations 



sin2J. _ sin 2B _ sin 2(7 



Pi Pi p 3 



are satisfied where p l} p 2 , p s are the three principal pitches of the three- 

 system. If the fundamental triangle has A, B, C for its angles then the 

 equation 



a, 2 sin 2 A + 2 2 sin IB + a/ sin 2(7 = 0, 



is the equation of the zero-pitch conic. It is however a well-known theorem 

 in conies that this equation represents a circle with its centre at the ortho- 

 centre, that is, the intersection of the perpendiculars from the vertices of the 

 triangle on its opposite sides. 



We thus have as the system of pitch conies 



a, 2 sin 24 + * 2 2 sin 2B + a 3 2 sin 2(7 - p (a, 2 + a 2 2 + a, 2 ) = 0. 

 B. 14 



