565] 
NOTE ON THE CARTESIAN. 
47 
respectively; this being so each of the Cartesians will, it is clear, pass through the 
point P 6 , and therefore also through P a and P c . 
The geometrical relations of the figure give 
to which might be joined 
aR 2 + ß S 2 + yT 2 = — aßy, 
aR' 2 4 ßS' 2 4 7T' 2 = — aßy, 
RT' 4 R'T = — ß(S + S'), 
ya = SS', 
yTT = aRR', 
R' 2 S 4 r (S 4 S') + R 2 S' = SS' (S + S'), 
T' 2 S + a 2 (S 4 S') 4 T 2 S' = SS' (S + S'), 
SR'T = S'RT, 
SP'R' = S'PR, 
but these are not required for the present purpose. 
As regards the first Cartesian, we have to verify that 
(ST + S'T) 2 (:TR' 4 T'R) 2 | (RS' - R'S) 2 _ Q 
a ^ 7 
The left-hand side is 
S 2 T' 2 + S' 2 T 2 + 2yoiTT' /3 2 (S 2 + S' 2 + 2ya) S 2 R' 2 4 S' 2 R 2 - 2yaRR' 
1 1 — 
a /3 y 
viz. this is 
= S> + /3 + ^ S ) + S' ! (~ + /3 + + 2a/3y + 2 (yTT - aRR), 
which is 
= S 2 (^f 2 ) + S' 2 ( Z ^) + 2^7 + 2 (yTT' - aRR'), 
/3 
and since the first and second terms are together = — 2 — S 2 S' 2 , that is, = — 2a/3y, 
ya 
the whole is as it should be = 0. 
In precisely the same manner we have 
(ST' - S'T) 2 (TR' 4- T'R) 2 | {RS' + R'S) 2 = Q 
a /3 7 
which is the condition for the second Cartesian: and the theorem in question is thus 
proved.