514] ON THE PROBLEM OF THE IN-AND-CIRCUMSCRIBED TRIANGLE. 
229 
21. I do not at present further consider the general theory, but proceed to con 
sider in order the 52 cases, interpolating in regard to the general theory such further 
discussion or explanation as may appear necessary. In the several instances in which 
the equation g = % + %' applicable, it is sufficient to write down the values of x> x > 
the mode of obtaining these being already explained. 
The 52 Gases for the in-and-circumscribed triangles. 
Case 1. No identities. 
X = BcDeFa, %' = FeDcBa (= x)> 
g = 2 aceBDF. 
Case 2. a — c — x. 
X = B (x — 1) DeFx, x = FeDxB {oc — 1) (= x)> 
g = 2x (x — 1) eBDF. 
Second process, for form a, = e = x. The equation of correspondence is here 
g ~ X ~ X + F ( e ~ 6 “ e ') = 0 ; 
but the points e being given as all the intersections of the curve a(=e) by the line- 
system cDe which does not pass through a, we have e — e — e' = 0 ; so that g = % + % ; 
and then 
x = BcDxF{x-1), % =F(x—l) DcBx, 
giving the former result^). 
Case 3. D —F = x. Reciprocation from 2; or else, second 'process, 
X = BcXe {X — 1) a, = Xe (X -1) cBa, 
g = 2X (X — 1) Bace. 
Third process: form F= B =x. We have here g = % + x ~~ Red. 
X = XcBeXa, = XeDcXa (= x ), 
X + X= %X 2 Dace; 
and the reductions are those of the first and second mode, as explained ante, Nos. 
11, 12, viz. each of these is = XDace, and together they are =2XDace\ whence the 
foregoing result. 
Case 4. a — D = x. 
X = BcXeFx, x — FeXBx (= %), 
g = 2Xx ceBF. 
1 Of course, the result is obtained in the form belonging to the new form of specification, viz. here it 
is = 2x (x - 1) cBDF; and so in other instances; but it is unnecessary to refer to this change.