# The Collected Mathematical Papers of Arthur Cayley Volume . 8

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This historic book may have numerous typos and missing text. Purchasers can usually download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1895 edition. Excerpt: ...to write down the values of x, x, the mode of obtaining these being already explained. The 52 Cases for the in-and-circwmscribed triangles. Case 1. No identities. x = BcDeFa, x'= FeDcBa (= x), g = 2aceBDF. Case 2. a = c = x. x = B(x-)DeFx, x' = FeDxB (x-1) (= x), g = 2x(x-)eBDF. Second process, for form a = e = x. The equation of correspondence is here g-XX+F(e-e-e') = 0; but the points e being given as all the intersections of the curve a(=e) by the linesystem cDe which does not pass through a, we have e--e--e' = 0; so that g = X + X; and then x = BcDxF(x-l), x' = F(x-l)DcBx, giving the former result). Case 3. D=F=x. Reciprocation from 2; or else, second process, x = BcXe(X-)a, X = Xe(X-l)cBa, g = 2X(X-l)Bace. Third process: form F= B=x. We have here g = X + X--Red. x-XcDeXa, = XeDcXa (= x), x + X'=2XDace; 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 (= x), g = 2XxceBF. 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. Observe this is what the result for Case 1 becomes on writing therein a = D=x, viz. the opposite curves a, D may become one and the same curve without any alteration in the form of the result. Case 5. a = B--x. X = (X-2)cDeFx, x' = FeDcX (x-2), where (X-2)x + X(x-2) = 2(Xx-X-x); therefore g = 2(Xx-X x)ceDF. Case 6. a = c = e = x: perhaps most easily by reciprocation of Case 7; or Second process, functionally by taking the curve a = c=e to be the aggregate curve x + at. The triangle aBcDeF is here in...show more

## Product details

• Paperback | 132 pages
• 189 x 246 x 7mm | 249g
• United States
• English
• black & white illustrations
• 1236837991
• 9781236837998