is 1, and that of its first, second, &c. principal minors are 1633. Prove that the determinant 1, cos a, cos (a + B), cos (a + B + y)| cos 8, cos (8+ a), cos (8+a+ẞ), 1 and all its first minors, will vanish when a +B+y+8=2π. 1634. Prove that, if u denote the determinant of the nth order, n+1 == 0; and thence (or otherwise) obtain its value in one of the three equivalent forms (2) sin (n + 1) a ÷ sin a, where 2 cos a = a, (3) (pr+1 − q′′+1) ÷ (p −q), where p, q are the roots of x2 - ax + 1 = 0. 1635. Prove that (1) |1-n, 1, 1, 10, n being the order of the determinant; ...... 1, 1-n, 1, ...... 1 1, 1, 1-n, 1 1, 1,.... (3)x,, 1, 1, 1, 2, 1, 1, 1, 23, ...... 1 1 1=P-P-2+2P-3-3P-4+...+(-1)"1 (n - 1), 1 where x, 21 1 ... x are roots of the equation x" - P1x"1 + P ̧x-9 — ... +(-1)"p. = 0; 1639. Prove that, if u denote the determinant of the nth order, น 41 + 1 = au + au- 1 --= 0; and express the developed determinant in the forms (1) "+1", -1, where v ̧ = (a − 1)* − (n − 1) (a − 1)*-' a-3 (a1)....... +1 (2) {p"+2-p"+1 - p − q2+2 + q*+1 + q} ÷ (p − q) (p + q − 2), where p, q are the roots of the equation x2 - (a− 1) x + 1 = 0; 1640. Prove that, if u denote the determinant of the nth order, ก -1 ƒ (x) = x′′ − a ̧x′′-1 + α ̧μ ̃ ̄2 — ...... + (− 1)'a ̧ = 0, is equal to 1; the second row being formed by differentiating the first with respect to x,, the third by differentiating the second with respect tox, and so on. DIFFERENTIAL CALCULUS. 1642. Having given sin x sin (a + x) sin (2a + x) ... sin {(n − 1) a + x} = 21-" sin nx, where n is a whole number and na: prove that (1) cot x + cot (a + x) + cot (2a + x) + (2) cot3x+cot' (a+x)+cot2 (2a +x) + ... +cot {(n − 1) a +x} = n cot nx, 1643. Prove that the limit of (cos x) cotax, as x tends to zero, is € ̄1. dy 1644. Prove that the equation (1-x) xy+1=0 is satisfied dx either by y√1-x=cos1x, or by y√x-1=log (x + √x2 − 1). 1645. Prove that the equation is satisfied by any one of the four functions C' (√1+x3±1)3, C′′ (√1+x2±x)3, and therefore by the sum of the four functions each with an arbitrary multiplier; and account for the apparent anomaly. |