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514] ON THE PROBLEM OF THE IN-AND-CIRCUMSCRIBED TRIANGLE. 
Third process. The equation of correspondence is 
g “ X ~ X + ^( e - 6 “ 6 ') = °> 
but for the correspondence (a, e) we have 
e —e—e'+_D(c—7 — y') = 0, 
and for the correspondence (a, c) we have 
c — 7 — y = BA, 
whence 
g=X + X+BDF. A; 
and then 
that is 
Moreover 
X = B(x-l)D(x-l)F(x-l), x' = B(x-l)D(x-l)B(x-l)(=x)l 
X + X = BDF. 2 (x - l) 3 . 
A=X — 2x + 2 + k 
(if k be the number of cusps of the curve a = c — e), and the resulting value is 
g = (2 (x — l) 3 + X — 2x + 2 + k) BDF; 
that is 
= {2x {x — 1) (x - 2) + X + k) BDF, 
where, however, the term kBDF is to be rejected. I cannot quite explain this; I 
should rather have expected a rejection = 2/cBDF, introducing the term — k. For 
consider a tangent from the curve D 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 draw B tangents cBa to the curve B, and F tangents eFa to the 
curve F; we have thus in respect of the given tangent of D, BF positions of a, or 
in all BDF positions of a which will ultimately coincide with the cusp; that is, BDF 
infinitesimal triangles of which the angles a, c, e coincide together at the cusp; and 
for all the cusps together kBDF such triangles: this would be what is wanted; the 
difficulty is that as (of the two intersections at the cusp) each in succession might 
be taken for c, and the other of them for e, it would seem that the foregoing number 
kBDF should be multiplied by 2. 
Case 7. B — D — F — x. Here g = % + X ~ anc ^ 
X — Xc(X — 1)e(X — 1) a, % ' = Ze (X-1) c(Z-1 )a(=%); 
that is, 
^ = 2X (X — l) 2 ace. 
The reductions of the two modes are as above, with only the variation that in the 
present case D is the same curve with the two curves B=F. That of the first mode is 
= X(X—1) ace, and that of the second mode is (2t + Si) ace, which is ={X(X—1)—x)ace: 
together they are = {2X (X — 1) — x] ace, or subtracting, we have 
g = {2X (X -1) (X - 2) + x) ace.