Transactions, Volume 13

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Page 98 - Euclid, but in which the famous postulate is assumed false, and in which the sum of the angles of a triangle is always less than two right angles.
Page 1 - ... with the algebraically integrable cases of this equation, we have rational and integral functions of s, derived from the polygon, the double pyramid, and the five regular solids, and which are called Polyhedral Functions. The Schwarzian Derivative occurs implicitly in Jacobi's differential equation of the third order for the modulus in the transformation of an elliptic function (Fund. Nova, 1829, p. 79) and in Kummer's fundamental equation for the transformation of a hypergeometric series (Kummer,...
Page 237 - The last figure is therefore in general correctly given to the nearest unit, but it may be in error by a unit where the tenth figure is a 4, 5 or 6.
Page 3 - ... had not been given by Schwarz, were obtained by Brioschi, and those of the remaining 3 cases (subject to a correction in one of them) were afterwards obtained by Klein. The first part of the present Memoir relates, say to the foregoing equation...
Page 256 - The numbers in parentheses denote the numbers of ciphers between the decimal point and the first significant figure; for example, in e""" there are 43 ciphers between the decimal point and the figures 136853947.
Page 261 - Iog10(e~z) from i to 500 at intervals of unity. The numbers in square brackets denote the numbers of figures between the last figure given and the decimal point ; and the numbers in parentheses denote the numbers of ciphers between the decimal point and the first significant figure. X...
Page 3 - ... and we have the theorem that the solution of this equation depends upon the determination of P, Q, R rational and integral functions of z, containing each of them multiple factors, which are such that P + Q + R = 0. Using accents to denote differentiation in regard to z, this implies P' + Q' + R = 0, and consequently QR - Q'R = RP - R'P = PQ - P'Q.
Page 1 - Z)\ then, if the equation of the second order be that of the hypergeometric series, generalised by a homographie transformation upon the variable x, the resulting differential equation of the third order is of the form...
Page 258 - ... the numbers of ciphers between the decimal point and the first significant figure. X bg.H <? e~x bg,„(e-*) 201 87-29319 08626 196 422 332 T).
Page 41 - NL : LM (L, M, N linear functions of z, x + iy, x—iy): showing that the relation between two stereographic projections of the same spherical figure is in fact that of a quadric transformation, the fundamental points in each figure being an arbitrary point and the two circular points at infinity: or, what is the same thing, to any line in the one figure there corresponds a circle in the other figure, which is the " circular relation

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