242 Journal of the Asiatic Society of Bengal. [April, 1908. 
2=me'; y=my’', 
From (1) a’ [m*(aa'* + Qha'y’ + by’*) + m(ga’ + fy’)] = 
—mzx'(mge' + mfy' + ©) 
miz'(ax'? + Zha'y’ + by’® + gx’ + fy’) = — ma'(ga’ + fy’ +6 
— (gx +fy'+e 
meat az’? + Qhar’y’ + by'®+ ga’ +fy 

. (3) 
— #' (gu' + fy’ +) 
ax’* + 2ha'y’ + by'* + ga’ + fy’ 
—y' (ge +fy' +0) 
aw’? + 2ha'y’ + by’* + ga’ +fy' / 


jes 
It is evident from (A) and (B) that the relation between the 
points is, in this case, a reciprocal one. For central curves . the 
second order, the point where the line jouiaig P to the origin 
meets the polar chord of P is the middle point of that shied the 
centre being the origin. 
The theorem stated in Art. 1, therefore, follows immediately. 
v§ a the two values of m given by the equations (2) 
and (3) we ge 
ax? + Zhay + by?+ ga+fy _ gu’ +fy' +e 
gatfyte ~ ax'® + Qha'y’ + by + gu’ + fy’ 
or (aa? + 2hary + by* + gu +fy) (aa’* + 2hzx’y’' + by’® + ga’ +fy’) 


=(gzr+fy+c)(ga'+fy’+c), 
so that if 8, 8 be the values of f(x, y) when we substitute in it 
the co-ordinates of a pair of inverse points and P, P’ those of the 
polar of the origin, the relation connecting two inverse points is 
(S—P)(8’— P')=PF’ (4), 
I have not, hitherto, been able to find any simple geometrical in- 
terpretation of this result. 

Space-analogues of the results of Arts. 6 and 7 can be 
ssa: obtain 
hus if fla, y, z)=90 represents the quadric 
ax’ + by* + c2z4 + Bfyz+ Qgzx + Shay + 2ux+ 2vy + 2wz4+d=0 
and (2’, y’, 2’, and 2, y, z) the co-ordinates of any point P and the 
point where the line joining P to the origin meets the polar plane 
of P with respect to the quadric, it is easily proved that 
