The theory of determinants in the historical order of development, by Sir Thomas Muir.

370 HISTORY OF THE THEORY OF DETERMINANTS and that therefore, if the determinant I \u1u221 vanish,* we must have U11 U 12 = 1 U21 %t22 bU2 and consequently cluI + c2u2 = 0. By way of illustration he takes u = 8x3-36x2y + 54xy2-27y3, and, finding the Hessian to vanish, obtains the relation 3u + 2u = 0, and 8u = 8(z —,y)3 To the work of Pasch, Gordan, and Noether he merely refers in a footnote. SYLVESTER, J. J. (1878): SHARP, W. J. C. (1879): ELLIOTT, E. B. (1881). [Questions 5762, 6077, 6756. Educ. Times, xxxii. p. 269, xxxiii. p. 60, xxxiv. p. 172; or Math. from Educ. Times, xxxiii. pp. 34-35, 92; xxxix. p. 41.] If u be a binary nthic it is known that 32u aD2u DU X0X2-+ ya = (n- l) and 2 a2 2 a2x y 2 y2U x2- + 2xy Dy+ + Y = n(n-1)u: and from these it readily follows that /Du\2 2'a 2 (n l)2x)- -n(n-l )a2 - y2H(u). This is the result numbered 5762 and attributed to Sylvester by Sharp when proving it.f It is, however, merely the simplest case * It might have been noted before this that the Hessian of u could be defined as the determinant of the set of equations got from applying to the first derivatives of u Euler's theorem regarding the differentiation of homogeneous functions. t The original ' question' bearing this number was proposed in September 1878, and is quite different from Sylvester's.

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Title
The theory of determinants in the historical order of development, by Sir Thomas Muir.
Author
Muir, Thomas, Sir, 1844-1934.
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Page 370
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London,: Macmillan and Co., Limited,
1906-
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Determinants

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"The theory of determinants in the historical order of development, by Sir Thomas Muir." In the digital collection University of Michigan Historical Math Collection. https://name.umdl.umich.edu/acm9350.0003.001. University of Michigan Library Digital Collections. Accessed June 20, 2025.
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