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1+i 1+i abd)P acd)P algebraical arguments coefficients coincident real compound partition conjugate contain convergence corresponding degree denote derived determinant distinct real divisible divisor doubly periodic function elements eliminating elliptic functions equal expression finite formula give given gnomonics H-functions H-T H-T Hence identical imaginary infinite integer l+2i l+2t linear relation logarithmic derivatives matrix modular equation modulus monomial multiplied negative notation nth roots obtained odd functions ordinary particular cognate forms points powers primitive quadric quadruple theta-functions quintic rational function real lines real plane real value relatively prime residues respect result root of unity rows and columns satisfied seminvariant sextic solution solvable squares substitution suffixes summation surds symmetric functions theorem theory theta-functions trinomial vanish vectors Weierstrass whence write zero
Page 115 - In one the object is to develop, in periodic series, certain functions of the moon's coordinates, which in number do not exceed five. This portion is the same whatever planet may be considered to act, and hence may be done once for all. In the other portion we seek the coefficients of certain terms in the periodic development of certain functions, five also in number, which involve the coordinates of the earth and planet only. And this part of the work is very similar to that in which the perturbations...
Page 282 - This method by which a matrix is robbed as it were of its areal dimensions and represented as a linear sum, first came under my notice incidentally in a communication made some time in the course of the last two years to the Mathematical Society of the Johns Hopkins University, by Mr CS Peirce, who, I presume, had been long familiar with its use. Each element of a matrix in this method is regarded as composed of an ordinary quantity and a symbol denoting its place, just as 1883 may be read 10 + 8h...
Page 2 - Now the meanings we have assigned to i, j, k are quite independent of, and not inconsistent with, those assigned to i, j, k. And it is superfluous to use two sets of characters when one will suffice. Hence it appears that...
Page 3 - X 1, 1 be taken as tlie unit of length, then the members of the equation have evidently not the same meaning, 1/1 being merely a numerical quantity, while 1 X 1 is a unit of area, it being a fundamental geometrical conception that the product of a length by a length Is an area, that of a length by an area a volume, while the ratio of two quantities of the same order as that of a length to a length Is a mere number of the order zero.
Page 95 - ... l)th degree. The proof applies substantially to each of the other groups. To prove the second part, it is only necessary to observe that, in the first of the groups (59), the last term is identical with the first, the last but one with the second, and so on. §56. Cor. 1. The reasoning in the proposition proceeds on the assumption that the prime number m is odd. Should m be even, the series A\ , 3i , etc., is reduced to its first term.