# A Binomial Congruence by Mike Swarbrick Jones

By Mike Swarbrick Jones

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Fc-0 fc=0 Equating the terms free of X and the coefficients of X, X2, . . , Xn-i, \n, we get ACo ACi ACi - Co - Cx = Uo I, = cti 7, = 02 /, ACn-i - c „_2 = 07J,—1 . - Cn—i = a* 7. Multiply these equations on the left by J, A, A 2, . . , A w_1, A71 respectively and add; we get 0 = a0/ + a iA + O2 A 2 + ••• + on_i A w 1 + anA n = ( A ). RANK OF MATRIX §26] 49 26. Rank of a matrix. Every matrix M having more than one element contains other matrices obtained from M by deleting cer­ tain rows or columns or both.

1) / = H ¿= 0 Consider a binary form ( ¿ ) Oi r, having prefixed binomial coefficients, as in §9. Under the trans­ formation Tn: x = %+ m7, y = let / become (2 ) f = i=0 where the A * are polynomials in n, do, ap whose actual ex­ pressions will not be needed. Differentiating (2 ) with respect to n, we get 0 = £ (? ) V* ~ A i ( p - i)| * -i- l i7i+1[ ’ since v = y is free of n, while f = x — m7. The total coefficient of v1' is © £ - ( A)<>-'+**»-* 24 ANNIHILATORS OF COVARIANTS §14] 25 in which the second term is to be suppressed when j = 0.

W e may start with (18),'p. 373, noting that u f, iif, and w f are O f, O f, and (0,0 — 0 0 )f of our text. 5Dickson, On Invariants and the Theory of Numbers, the Madison Colloquium of the Amer. Math. , 1914. Cf. Glenn, Treatise on the Theory of Invariants, 1915, 175-208. Dickson, History of the Theory of Numbers, III, 1923, 293-301. 6Wilczynski, Proc. National Acad. Sciences, 4, 1918, 300-5. C hapter II I MATRICES, BILINEAR FORMS, LINEAR EQUATIONS Chapters III-V I, which are independent of I-I I, give a new exposition of the subject usually called higher algebra.