THE IN-AND-CIRCUMSCRIBED TRIANGLE. 
409 
which is the solution : the value obtained above was b = /+;s, and we in fact have iden- 
tically 
— 3x — 3X + 2§. 
It was in this manner that I originally applied the principle of correspondence to inves- 
tigating the number of inflections of a curve, regarding, however, the term x as a special 
solution ; it is better to put the cusp and inflection on the same footing as above. 
29. Correspondence [a, c ). 
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)0— 3)+(X— 6)(— 2#— 2X+2 + £) 
= — . 2X 2 + 8X+ + (X — 6)s ; 
this is in fact=2r + (X — 3)&, viz. we have 
2r=X 2 — X + &r — 3| 
(X-3)*=(X-3)(-3X+|)=-3X 2 +9X + (X-3)|, 
and therefore 
2r+(X-— 30=as above, 
viz. the united points ( a , c ) are the 2r points of contact of the double tangents, and the 
x 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. 
c 0 To To — 0 6 ) 
c 0 =— 2r 3 + 8#+SX + (;r— 6)| . 
= 2c$ +0—3)/. 
31. It maybe remarked, as regards the cases which follow, that although the result in 
terms of (ci, x , /, r) 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 (x, X, |), as given by the prin- 
ciple of correspondence, to the required form in terms of (&, x, /, r) 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' 0 and b — /3 — /3' we have 
d-+ — — (2X+2#— 18)A, 
3 L 
MDCCCLXXI. 
