PROFESSOR CAYLEY ON POLYZOMAL CURVES. 81 



systems of values of p : <r : t. To verify this, observe that writing the equation 

 under the form 



the second equation is verified ; and that writing them under the form 



s : a : r = - (/3 + 7) (a + 3) + M : - (7 + a) (0 + 5) + M : - (a + /3) (7 + 5 ) + M , 



where 



M = 187 + ah + ya -f /i5 + a/3 + 75 — 2 sja(3y8, 



the second equation is also verified. 



185. If we assume for a momenta = cos a + i sin a — e ia , &c, viz., if a, b, c, d 

 be the inclinations to any fixed line of the radii through A,B,C,D respectively, 

 then we have 



J*b± Jfyy =e i(a + b + c + d)i {«*<« + «-»-* ±e -i(a + d- b -oii > 

 Ja(j3 - 7) =e* (0 + *+ e) ' { e *<»-«>•' _ 6 -i(&-<0« |, 



and thence 



^/ag (/3 — 7) : \//3rf(7 — a) : ^r (a — /3) = cos 4 (« f d — & — c) sin £ ( ft — c) 



: cos ]; (b + d — c — a) sin £ ( c — a) 

 : cos 4 (a + rf — a — b) sin -|- (a — &) ; 



or else = sin £ (« + d — -b — c) sin \ {b — c) 



: sin -} (6 + rf — c — a) sin £ (c — a) 

 : sin -] ( c + d — a — ft) sin \ (a — b) . 

 Putting in these formulae, 



\{a — b — c) — A, then we have B — C = £(& — c) , 

 |(& — c — a) = 5, „ C — A=l(c — a) , 



i( c _ a _ h) = C, „ 4 - J?= }(a - 6) , 



and for either set of values the verification of the relation 



isAg(j8 - 7) + VM7 - «) + VtK" - (3) = , 

 will depend on the two identical equations 



sin A sin (B — C) + sin 5 sin (C — ^) + sin Csin (^4. — B) = , 

 cos A sin(B — C) + cos i? sin (C — A) + cos C sin (A — B) = : 



although the foregoing solution for the case of a circular cubic is the most elegant 

 one, I will presently return to the question and give the solution in a different 

 form. 



Focal Formulce for the Symmetrical Curve — Art. No. 186. 



186. In the symmetrical case, where the foci A, B, C,D are on a line, then 

 if, as usual, a, b, c, d denote the distances from a fixed point, we have the ex- 



VOL. XXV. PART I. X 



