Vol. IV, No. 4.] Reciprocal Relations of Qurves and Surfaces, 241 
(N.S.] 

5. Generally, if 2’, y’ be the co-ordinates of any point P in 
the plane of the curve Ha, y)=0 and X, Y those of the point 
where the line joining P to the setts meets the first ae ar of the 
curve with respect to P, and if the equation of the curve be ren- 
dered homogeneous by ‘the introduction of the aces os un it z, it is 
easy to prove that 
id Pi 
, “dz 7 : d 
| af, af’ ae 
oa oe ap 
i ag "ae Vd 
It is easily seen that the ren between the two points 2’, y’ and 
#, y is not, in general, reciproc 
the particular case, however, Minne f(#, y) =0 repre- 
sila: the cipal. equation of the second degr 
ax* + Qhaey + by®+2ge+2fy+c=0 we have 

____ (ge +fyte) yf gat sod 2s A Soa ge 
ax? + Zhay + by® + ga + fy!” ~ aa + Qhay + by? + gat fy 
or x’ (aa®+ Lhay + by? + gx + fy) = ~ alga — Meee (1) 
y' (aa® + Qhay + by® + gx +fy) = —y(getfy +e) 
aad + Qhay + by? + ge fy _ M (SAY) ..cerers » (2) 

whence — = - = 
@ say gut fyt+e 
