Val. IV, No. 4.) Reciprocal Relations of Curves and Surfaces. 243 
N.S. 


wy aaah + by* +24 + Ofyz + 2gzw + Shay + uw+ vy + we 
oe eae a ux +vy+wzt+d 
from which it can be proved as in Art. 6, that 
e y 8 Ge rby* teeF+ 2... tre oer 
ef ¢ ux’ + vy’+w2' 4d 
whence it is evident that the relation meer x’, y'’, 2 and ay, 2 
is reciprocal. 
We have further the identity 
—P) (S’—P’)=PP’ exactly analogous to that of Art. 7 
9. When f(a, y, fig represents the surface 
ye ot 




a 
; x roe a’ 
C= an = eee 
zg yf gt ge ye 7 
atyta at pte 
v ¥ be y' 
1~_ yy 2 y= ya yt zi 
at pte at ata 
: Z es ? 
z P a “ ~ 2s y? 12 S 
a Bte a > pt 
So that if the point 2’, y’, 2’ be constrained to move on . the surface 
F(#, y, z) =0, the inverse dpi moves on the surface 


: . os 0 (5) 
aig gs — 2 Pi y* a) rege % 
4 Rta atpta atyte | 
It is evident from the above expressions for 2’, y’, 2’ in terms 
of 2, y, z and vice versa that the agore x, y z is the centre of the 
polar plane of the point x’, y’, 2 with respect to the surface 
athe oe a nia a 234, Frost’s Solid Geometry) and 
vice versa. os 
As an interesting panies case of 5) let F km Ys ice 0 
2. 7% 
_represent the conicoid = = < +55”. 
a = pa 
