Sentence Examples


  • Since the determinant having two identical rows, and an3 an3 ��� ann vanishes identically; we have by development according to the elements of the first row a21Au+a22Al2 +a23A13+��� +a2nAin =0; and, in general, since a11A11+a12A12 +ai 3A13+�� � +ainAin = A, if we suppose the P h and k th rows identical a A +ak2 A 12 +ak3A13+��� +aknAin =0 (k > i) .and proceeding by columns instead of rows, a li A lk +a21A2k + a 31A3k+���+aniAnk = 0 (k .>
  • Since A lk is a determinant we similarly obtain Alk = a21Alk+� � � +a2,k-iAl,k +a2,k+lAl,k+ ���+a2 21 2,k-1 2, k +1 2,n and thence = Xalia2kAli where k; i,k 2k and as before A = a1, an A i> k i,k I ail, auk 12k an important expansion of A.
  • The Augustan age was one of those great eras in the world like the era succeeding the Persian War in Greece, the Elizabethan age in England, and the beginning of the 19th Lk y.
  • The formula, in the above case, is 3h{ *k(uo,o + 3 where u 0, 0 denotes the ordinate for which x=xo+Oh, y=yo+c¢k The result is the same as if we multiplied lk(vo 3v1+3v2 + v 3) by lh(uo 4u1 +u2), and then replaced uovo, uov1, ..
  • The above is a particular case of a general principle that the obtaining of an expression such as Ih(uo+4u1+u2) or lk(vo-1-3v 1 +3v 2 +v 3) is an operation performed on uo or vo, and that this operation is the suns of a number of operations such as that which obtains 3huo or 1kvo.

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