THE IN-AND-CIRCTJMSCRIBED TRIANGLE. 
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And the second-mode reduction is 
«(2r+3/)c(c— 1) 
(where r, / refer to the curve B = D = F), which is 
that is, the reduction is =ay{y— 1){X(X— 1)— a}. 
Hence the two together are —ay(y — 1){2X(X — 1)— a}; and subtracting from ^ -f- yj 
we have 
g=ay(y- 1). {2X(X-l)(X-2)+a}; 
but on account of the symmetry each triangle is reckoned twice, and the number of 
triangles is =-|g. 
Case 29. a—c— B=a, D=F=y. 
X— (X— 2)(a— 8)Y e(Y— l)a, ^=Y«(Y-l>r(X-2)(X-3)(=: X ), 
g=2a(a-3)(X-2)Y(Y-l>. 
Case 29. Second process. Taking the form 
C=D=e=i', B— F— ^ ; 
here 
n o-= 5C+^'— Red -> 
% =Ya(X-2)(a-3)Y a, =y>, 
and 
% +x'=2Y 2 a(a-3)(X-2)«. 
There is a first-mode reduction, 
aY{2r+2S(X — 4) + 3*(X -3)}, 
viz. this is 
aY { X 2 -X+8a-3£ 
+ (X — 4)(a 2 —x-\- 8X — 3|) 
+ (X— 3)( -9X+3§)}, 
which is 
= aY { X(a 2 — a — 6) — 4a’ 2 + 1 2a} ; 
and a second-mode reduction, 
«YX(a— 2) (a— 3). 
Hence the two together are 
= «Y{X(2a 2 — 6a) - 4a 2 + 12a} 
=2Ya(a-3)(X-2 )a, 
whence the result is 
= 2(Y 2 — Y)a(a— 3)(X— 2)«, 
which agrees with that obtained above. 
On account of the symmetry we must divide by 2. 
