Calendar: 1874-1875 Page 551

GENERAL LITERATURE AND SCIENCE 551 that it will consist of an equal number of white and black balls 11 Show how to express given number whether integral or fractional in any scale of notation Express 142 in the scale whose radix is 12 An equation cannot have more positive roots than fix" has changes ot sign of successive coefficients nor more negative roots than has changes of sign 13 If be the numerical value of the greatest negative coefficient in the equation JDj then ρ is superior limit of the positive roots of the equation 14 If α β γ be roots of the equation x3 px prove that the equation of which the roots are 3y β Ύ β Φ px χ 15 Enunciate Sturm's Theorem and apply it to find the number and position of the real roots of 2x 199 Plane anK pljcrical Crioctnotnetn Show how to express sin and cos in terms of sin Λ Give priori reasons for the four values Prove the formulae- sin2 2a in 2β ta β ta β eot2 cosec cosec2 cosec2 coscc3yi cosecy cot C013A One of the angles of plane triangle is 60 and the sides including it are in the ratio of show that the other angles are -1 V3 -1 π tan -ζ- ana tan υ