254 
ON THE PROBLEM OF THE IN-AND-CIRCUMSCRIBED TRIANGLE. 
[514 
29. Correspondence {a, 6). 
Since b — /3 — ¡3' — 2A, we have here 
c — y — y = (X — 6) A, 
and 
y — y — (X — 2) (x — 3), 
whence 
c = 2 (X - 2) (x - 3) + (X - 6) (- 2x - 2X + 2 + £) 
= - 2X 2 + 8X + 8x + (X - 6) £; 
this is in fact =2r + (X — 3) k, viz. we have 
2t= X 2 — X + 8x — 3f 
(X-3)K = (X-3)(- 3X+^) = -3X 2 + 9X + (X-3)l 
and therefore 
2t + (X — 3) k = as above, 
viz. the united points (a, c) are the 2t points of contact of the double tangents, and 
the k cusps each (X — 3) times in respect of the (X — 3) tangents from it to the 
curve. This is the way in which I originally applied the principle to finding the 
number of double tangents of a curve. 
30. Correspondence (B, D). By reciprocation 
Co - To - Yo' = (« - 6) A, 
c 0 = — 2x 2 + 8x + 8X + {x — 6) £ 
= 28 +0-3) i. 
31. It may be remarked, as regards the cases which follow, that although the 
result in terms of (8, k, l, t) when once known can be explained and verified easily 
enough, there is great risk of oversight if we endeavour to find it in the first 
instance; while on the other hand the transformation from the form in terms of 
0, X, £), as given by the principle of correspondence, to the required form in terms 
of (8, k, i, t) is by no means easy. I in fact first obtained the expression in (x, X, £), 
and then, knowing in some measure the form of the other expression, was able to 
find it by the actual transformation of the expression in (x, X, £). 
32. Correspondence (a, D). 
From the values of c 0 — y 0 — y/ and b — /3-/3' we have 
d - 8 - 8' = - (2X + 2x - 18) A, 
and then 
8 = (X — 2) (x — 3) (X — 3), 8 / = («-2)(X-3)(a;-3), 
whence 
d = (x — 3) (X — 3) (X + x — 4) 
+ (- 2X - 2x + 18) (- 2X - 2x + 2 + f)