THE IN-AND-CIRCUMSCRIBED TRIANGLE. 
387 
it is easy to see that a, (3, y are independent of the curves B, D, F ; and taking each of 
these to be a point, and the curve a=c=e to be a conic, then it is known that <px—2, 
we have 
2 = 16 — 24 + 2a + 2/3+ Gy, 
that is 
a+j3 + 3y=5. 
The case where the curve a=c=e is a line gives 0 = 2 — 6+a+3y, that is, 
a + 3y=4 ; 
but it is not easy to find another condition; assuming however y=0, we have a=4, 
/3 = 1, and thence 
<px=(2x 3 — 6^ 2 +4^r+X)BDF, 
or say 
g={2^-l)(a:-2)+X}BDF: 
this is a good easy example of the functional process, the use of which begins to exhibit 
itself ; and I have therefore given it, notwithstanding the difficulty as to the complete 
determination of the constants. 
Tim'd process. The equation of correspondence is 
g-%-%'+F(e-£-s')=°, 
but for the correspondence (a, e) we have 
e — z — £ + D(c — y — y') = 0, 
and for the correspondence (a, c) we have 
c — y — y'=BA, 
g=X+x'+ BI) F. A ; 
*=B(*-l)D(*-l)F(ff-l), ^=F(^-1)D(^-1)B^-1)(= Z ) ; 
X + ^ = BI1F . 2 (^ — 1 ) 3 . 
A = A — 2# + 2 + z 
(if k be the number of cusps of the curve a—c=e), and the resulting value is 
g={2(*-l) 3 +X-2.£+2+*}BDF; 
= { 2x(x - 1)(^- 2) +X+*} BDF, 
whence 
and then 
that is 
Moreover 
that is 
where, however, the term ^BDF is to be rejected. I cannot quite explain this ; I should 
rather have expected a rejection =2^BDF, introducing the term —x. For consider a 
tangent from the curve I) from a cusp of the curve a—c=e : there are D such tangents ; 
each gives in the neighbourhood of the cusp two points, say c, e ; and from these we 
