Relations among the minors of an evanes cent determinant. Case of equal roots. 343 c. When there are equalities among the roots the problem has generally solutions of the form 41 = (c1t + b1) e^t, ¥2 = (c ̧t+b.) eλt, etc. ..(6). To prove this let A, X' be two unequal roots which become equal with some slight change of the values of some or all of the given constants (11), 11, 11, (12), 12, 12, etc.; and let 1 = A1'ext - A1et, 2 = Â1⁄2'é1 ́t - Aet, etc...(6)' be a particular solution of (1) corresponding to these roots. Now let To find proper equations for the relations among b1, b2, . . . C1, C2, . . . in order that (6)"" may be a solution of (1), proceed thus :-first write down equations (3) for the X' solution, with constants A ̧,4,', etc.: then subtract from these the corresponding equations for the A solution: thus, and introducing the notation (6)”, we find {(11) X'2 + 11X' + 1 1 } c, + {(12) X ́2 + 12X' + 12} c2+ etc. 01 {(21) X' +21X' +21} c, + {(22)2 + 22X' + 22}.c, + etc. = 0...(6), {(11) X'2 + 11X' + 1 1 } 6, + {( 1 2 ) λ'2 + 12X' + 12} b, + etc. 2 = — [c2 — b ̧ (λ' ́ — λ)] {(1 2) (A ́+ X') + 12} – etc. etc. - - [c2 — b ̧ (λ' - λ)] {(22) (λ ́+λ') + 22} + etc. etc. ...(6). Equations (6) require that A' be a root of the determinant, and i-1 of them determine i-1 of the quantities c1, c2, etc. in terms of one of them assumed arbitrarily. Supposing now c1, Ca, etc. to be thus all known, the i equations (6) fail to determine the i quantities b, ba, etc. in terms of the right-hand members because X' is a root of the determinant. The two sets of equations (6) and (6) require that A be also a root of the determinant and i - 1 of the equations (6)' determine i - 1 of the quantities b1, b,, etc. in terms of c,, c, etc. (supposed already Case of known as above) and a properly assumed value of one of the b's. 343 d. When X' is infinitely nearly equal to λ, (6)"" becomes infinitely nearly the same as (6), and (6) and (6) become in terms of the notation (5) I'I C + I2 C2+ etc. = equal roots. vii These, (6), (6), are clearly the equations which we find simply by trying if (6) is a solution of (1). (6) requires that A be a root of the determinant D; and they give by (5)" with c substituted for a the values of i-1 of the quantities c1, c, etc. in terms of one of them assumed arbitrarily. And by the way we have found them we know that (6) superadded to (6) shows that A must be a dual root of the determinant. To verify this multiply the first of them by M(11), the second by M (2.1), etc., and add. The coefficients of b2, b1, etc. in the sum are each identically zero in virtue of the elementary constitution of determinants, and the coefficient of b1 is the major determinant D. Thus irrespectively of the value of A we find in the first place, Using successively the several expressions given by (5) for these ratios, in (6)vil, and putting D = 0, we find which with D = 0 shows that A is a double root. Suppose now that one of the c's has been assumed, and the others found by (6): let one of the b's be assumed the other Case of equal roots. i - 1 b's are to be calculated by i-1 of the equations (6). Thus for example take b1 = 0. In the first place use all except the first of equations (6) to determine b, b, etc.: find d2 1 == M(x+1)b,=={M(1,21,2) "{'x' + M(1,21,3) "/" - {M(1, 2*1, 2) we thus αλ M(11)b, etc. M(11)b, etc. etc. Secondly, use all except the second of (6)TM" to find b, b1⁄2, etc. : Case of equal roots and evanescent minors: vil Thirdly, by using all of (6) except the third, fourthly, all except the fourth, and so on, we find M (31) b1 = etc., M(31) b2 = etc., M (31) b = etc....... (6) 1. 343 e. In certain cases of equality among the roots (343 m) it is found that values of the coefficients (11), 11, 11, etc. differing infinitely little from particular values which give the equality give values of a, and a,', a, and a', etc., which are not infinitely nearly equal. In such cases we see by (6)" that b1, b2, etc. are finite, and c,, c2, etc. vanish: and so the solution does not contain terms of the form tet: but the requisite number of arbitrary constants is made up by a proper degree of indeterminateness in the residuary equations for the ratios b/b1, b/b1, etc. Now when c1 = 0, c, 0, etc. the second members of equations (6)x, (6)x, (6), etc. all vanish, and as b, b1, b1, etc. do not all vanish, it follows that we have M(1·1) = 0, M (2·1) = 0, M (3·1) = 0, etc........(6)xil. Hence by (5) or (5)vii we infer that all the first minors are zero for any value of A which is doubly a root, and which yet does not give terms of the form tet in the solution. This important proposition is due to Routh*, who, escaping the errors of previous writers (§ 343 m below), first gave the complete theory of equal roots of the determinant in cycloidal motion. * Stability of Motion (Adams Prize Essay for 1877), chap. 1. § 5. theorem. He also remarked that the factor t does not necessarily imply Routh's motional 343 f. We fall back on the case of no motional forces by Case of no taking 110, 120, etc., which reduces the equations (3) for forces. determining the ratios a,/a,, a/a,, etc. to + 11a, + 12a ̧+ etc. = 0, da or, expanded, 01 [(11)2+11]a, +[(12) λ2 + 12] a,+ etc. =0....(7). [(21) X2 + 2 1 ] α, + [(22) λ2 + 22] a ̧ + etc. = 0 S 2 The determinantal equation (4) to harmonize these simplified equations (7) or (7) becomes This is of degree i, in X: therefore A has i pairs of oppositely signed equal values, which we may now denote by and for each of these pairs the series of ratio-equations (7') are ¥1 = (Aeλ+Bɛ−λ) + (A'ext + B'e-X't) + a" (A"ext+B'e "t) + etc. a a απ ' (Ae^t + Be−λ) + "', (A'ext + B'e-xt) + ",, (A"ext + B" e-X"() + etc. etc. where A, B; A', B'; A", B"; etc. denote 2i arbitrary constants, are i sets of i-1 ratios each, the values of which, when all the (9), Case of no motional forces. Cycloidal motion. Conservative positional, and no motional, forces. 343 g. When there are equal roots, the solution is to be completed according to § 343 d or e, as the case may be. The case of a conservative system (343 h) necessarily falls under § 343 e, as is proved in § 343 m. The same form, (9), still represents the complete solutions when there are equalities among the roots, but with changed conditions as to arbitrariness of the elements appearing in it. Suppose λ= '" for example. In this case any value may be chosen arbitrarily for a,/a,, and the remainder of the set a/a,, a/a, ... are then fully determined by (7'); again another value may be chosen for a/a, and with it a,'/a,', a'a', ... are determined by a fresh application of (7′) with the same value for A: and the arbitraries now are A+ A', B + B', C2 A + a a2 A', a1⁄2 B+a1⁄2, numbering still 2i in all. Similarly we see how, beginning 343 h. For the case of a conservative system, that is to say, the case in which 12=21, 13=31, 23=32, etc., etc.............(10), the differential equations of motion, (1), become and the solving linear algebraic equations, (3), become .(10), .(10"), V=1(114, +2.124,12+etc.), and =}(11a,3+2. 12a,a,+ etc.)... (10′′). In this case the i roots, X2, of the determinantal equation are the negatives of the values of a, ß, ... of our first investigation; and thus in (10"), (8), and (9) we have the promised solution by one completely expressed process. From § 337 and its footnote we infer that in the present case the roots A are all real, whether negative or positive. |