Calendar: 1875-1876 Page 580

576 GENERAL LITERATURE AND SCIENCE and the value of the expression Ι ί Ι ί In how many trials is it an even wager to throw sixes with two dice Define the root of an algebraical equation Assuming that every equation has root shew that an equation of the th degree has roots and no more If two numbers when substituted for give results with different signs then some odd number of roots lies between them if they give results affected with the same sign then an even number of roots or no root lies between them If be divisor of the last term of fix find criteria to be satisfied in order that may be an integral root of the equation and solve Xs 13 16 48 10 The roots of reciprocal equation of an even number of dimensions exceeding quadratic may be found by the solution of an equation of half the number of dimensions Ex 11 Prove that the cubic a3 px2 -f qx may be reduced to the form ay cf by the substitution and that the roots of the quadratic for determining are α β β γ י β Γ Ά'β β fl yf γ ך β α β γ being the roots of the original equation 12 State Sturm's Theorem and indicate the successive steps of the proof Prove that the equation i3 has two roots between and and one root between and 13 Prove by Horner's method that the root of the equation Xs -0 71 42 ץ which lies between and is 5'990176