PROFESSOR CAYLEY ON POLYZOMAL CURVES. 



39 



CA, BD in S, and the lines AB, CD in Tj we may, to fix the ideas, consider 

 A BCD as forming a convex quadrilateral, R and T will then be the exterior 

 centres, S the interior centre ; a, b, c, d, may be taken equal to BCD, — CD A, 

 DAB, — ABC, where the areas BCD, &c, are each taken positively. The 

 expression aA + bB + cC + dD has the same value, whatever is the position 

 of the point P {x, y, z = 1) ; taking this point at B, and writing for a moment 



BA = a, RB = /3,BC = y, BD = h, 

 then 



BCD = (ROD - BBD) = ±RD (RC - BB) sin B = (y-fyd sin B , 



with similar expressions for the other triangles ; and we thus have 



aA + bB + cC + dD = W . sini2 



) -j3\d-a)y[_ 

 1 + 7 2 (d-a)/3 



£z 2 sin R(fiy - ttij(y - /3;(3 - a) 



that is, replacing a, /3, y, S, by their values, and writing also 0=1, we have 



aA + bB + cC + dD = J sinij . (RB . BO - BA . BD)BG . AD , 



where ^sin R.BC.AD is in fact the area of the quadrilateral ABCDj we have 



thus 



aA + bB + cC + dD = (RB.BC -BA.RD)U 



= (SG .SA -SB.SD)U 



= (TA.TB -TC.TD)U 

 where it is to be observed that SA, SC being measured in opposite directions 

 from S, must be considered, one as positive, the other as negative, and the like as 

 regards SB, SD. This expression for the value of the constant is due to Mr 

 Crofton. In the particular case where A, B, C, D, are on a circle, we have as 



before 



aA + bB + cC + dD = . 



89. If the four points A,B,C, D, are on a circle, then, taking as origin the 

 centre of this circle and its radius as unity, the circular co-ordinates of the four 

 points will be 



the corresponding forms of A°, &c, being 



A° = (g - ca) (n - -z) - a" 2 z 2 , &c. 

 the expressions for a, b, c, d, observing that we have 







l 





0,|3- 



,i 



= fyh 



1, ft /3 2 



y>y 



M 



!> y> 7 2 



i, i- 



M 





1, 8, a a 



I 



