8x 0=m w. 8 H2 m d x m dt dn m m dt on 8 H, 2 m 88 , the six quantities x a po v T w being constant for the undisturbed motion of any one binary system ; and therefore the six functions x{1), 7(2), (3), (4), 4(5), %(6), or x, n, this V, 7, w, being such as to satisfy identically the following equation, 8 x 8 H, 8 x 8 H 8 x 8 H δκαι δ Η, 8 x 8 H δκαι δ Η + (L2) १० with five other equations analogous, for the five other elements ng fu, v, T, w, in any one binary system (m, M). 33. Returning now to the original multiple system, we may retain as definitions the equations (K2.), but then we can no longer consider the elements x; ; fH; Y; T; W; of . ; the binary system (m, M) as constant, because this system is now disturbed by the other masses m,; however, the 6 n- 6 equations of disturbed relative motion, when put under the forms de δ Η, 8 HZ ६. + (M2.) + " and combined with the identical equations of the kind (L?.), give the following simple expression for the differential of the element x, in its disturbed and variable state, δκαι δ Η, δκαι δ Η, 8x HQ 8 x 8 8 x 8 H, 8 x 8 H, + 8x & Ę dt N2. on sy dyl on 8 88 82 8 z 88 together with analogous expressions for the differentials of the other elements. And if we express & n & ac y' z', and therefore H, itself, as depending on the time and on these varying elements, we may transform the 6 n - 6 differential equations of the 1st order, (M2.), between $ ng " y' z' t, into the same number of equations of the same order between the varying elements and the time, which will be of the forms 8 H, 8 H, 7 8 W 8 H, 8 H, + {a, w} 8 W 8 + {pu, v) HA + {2, 7) Hi + {y, a) m {fi, x} + {fig a } + f, + w} 8 8 H (02.) + {v, a} W + {v, kes m = {v, x} , a {v, 7) SH+ {v, w + } v} 8 H = {1,2} T dt } T w 8 H, +{w, v} 7 dt δλ X X . if we put, for abridgement, X 8 x 8 8 x 8 a + (P2) and form the other symbols {x,y}, {2, x},&c., from this, by interchanging the letters. It is evident that these symbols have the properties, {2, x} = - {x,a}, {x, x} = 0; . ) (184.) and it results from the principles of the 15th number, that these combinations {x, 2}, &c., when expressed as functions of the elements, do not contain the time explicitly. There are in general, by (184.), only 15 such distinct combinations for each of the n-1 binary systems; but there would thus be, in all, 15 n- – 15, if they admitted of no further reductions : however, it results from the principles of the 16th number, that 12 n – 12 of these combinations may be made to vanish by a suitable choice of the elements. The following is another way of effecting as great a simplification, at least for that extensive class of cases in which the undisturbed distance between the two points of each binary system (m, M) admits of a minimum value. الح . Simplification of the Differential Expressions by a suitable choice of the Elements. 34. When the undisturbed distance r of m from M admits of such a minimum q, corresponding to a time t, and satisfying at that time the conditions pd = 0, m!>0, (185.) then the integrals of the group (12.), or the known rules of the undisturbed motion of m about M, may be presented in the following manner : x= v{(ty' - 92')2 + (12' – $)2 + (SX' – $7')2} ; M+m (x'2 + y2 + x2) – Mf(r); (Q2.) ; X . м dr M + mo n d gues M + m dr M 2 + 2 Mf(r) – (1+ ( Mge? the minimum distance q being a function of the two elements x, y, whic! must satisfy the conditions >0; qe M qo and sin-'s, tan-'t, being used (according to Sir John Herschel's notation) to ex 24 +2 M f (9) - (1 + m) = 0, M f" (9) + (1 + m) >0; . (186.) dr press, not the cosecant and cotangent, but the inverse functions corresponding to sine and cosine, or the arcs which are more commonly called arc (sin = 8), arc (tan= t). s. It must also be observed that the factor which we have introduced under the Nd 7.29 signs of integration, is not superfluous, but is designed to be taken as equal to positive or negative unity, according as dr is positive or negative ; that is, according as r is increasing or diminishing, so as to make the element under each integral sign constantly positive. In general, it appears to be a useful rule, though not always followed by analysts, to employ the real radical symbol VĒ only for positive quantities, unless the negative sign be expressly prefixed ; and then to will denote posi N22 tive or negative unity, according as r is positive or negative. The arc given by its sine, in the expression of the element w, is supposed to be so chosen as to increase continually with the time. 35. After these remarks on the notation, let us apply the formula (P2.) to calculate the values of the 15 combinations such as {x,7}, of the 6 constants or elements (Q.). Since r= v (52 + y2 + 32), . (187.) it is easy to perceive that the six combinations of the 4 first elements are as follows: {2,4} = 0, {x,y} = 0, {x,v} = 0, {n, k} = 0, {2, v} = 1, {fu, v} = 0. . (188.) To form the 4 combinations of these 4 first elements with 7, we may observe, that this 5th element 7, as expressed in (Q2.), involves explicitly (besides the time) the distance r, and the two elements , fu; but the combinations already determined show that these two elements may be treated as constant in forming the four combinations now sought; we need only attend, therefore, to the variation of r, and if we interpret by the rule (P2.) the symbols {x, r} {1, r} {f, r} {v, r}, and attend to the equations (I?.), we see that dr {n, r} = 0, {, r} = 0, {fi, r} = (189.) dr being the total differential coefficient of r in the undisturbed motion, as determined by the equations (12.); and, therefore, that {x,7} = 0, {9,7} = 0, {v,7} = 0, (190.) and &r dr dt dr (191.) observing that in differentiating the expressions of the elements (Q2.), we may treat those elements as constant, if we change the differentials of Šn Sac y' x' to their undisturbed values. It remains to calculate the 5 combinations of these 5 elements with the last element w; which is given by (Q2.) as a function of the distance r, the coordinate X, and the 4 elements a, a, M, v; so that we may employ this formula, 8 W {e, w} = p {e,r} + {e,8} + {e, x} + {e, a} + mu se, p} + {e, v}, (192.) + e {f e,}, ) = dt, {v,r} = 0, ť dt Ô w 8 w 8 δη 8 w δκαι δλ M in which, if e be any of the first five elements, or the distancer, de de ,} 8} ,x, ox and (194.) w dζ δω or dr 8 g the formula (192.) may therefore be thus written: & e + in + & , {e, w} = du 8z de & ad tny' +82 da (195.) 8 {e,a} + δλ δμ {e, }. at ou ; We easily find, by this formula, that drew (196.) and 8y w 8 w (197.) The formula (195.) extends to the combination {7, w} also; but in calculating this last combination we are to remember that r is given by (Q2.) as a function of x, l, r, such that δλ and thus we see, with the help of the combinations (196.) already determined, that Tw 8 x *® (199.) ou if we represent for abridgement by ®, and 12, the coefficients of dr under the integral signs in (Q2.), namely, M + m (200.) M 80 8.12 (201.) r, being any quantity which does not vary with the elements x and y; we might therefore at once conclude by (199.) that the combination {T,w} vanishes, if a diffi MDCCCXxxv. T culty were not occasioned by the necessity of varying the lower limit 92 which depends on those two elements, and by the circumstance that at this lower limit the coefficients , 12, become infinite. However, the relation (202.) shows that we may express this combination {7, w} as follows: {7, w} = S," O, dr + S," 2, dr, (203.) r, being an auxiliary and arbitrary quantity, which cannot really affect the result, but may be made to facilitate the calculation; or in other words, we may assign to the distance r any arbitrary value, not varying for infinitesimal variations of r, lig which may assist in calculating the value of the expression (199.). We may therefore suppose that the increase of distance r - q is small, and corresponds to a small 9 positive interval of time t - 7, during which the distance r and its differential coefficient r' are constantly increasing; and then after the first moment 7, the quantity 1 (204.) I will be constantly finite, positive, and decreasing, during the same interval, so that its integral must be greater than if it had constantly its final value; that is, = 5;"0,dr> (r – 9) or (205.) Hence, although O, tends to infinity, yet (r — 9) ©, tends to zero, when by diminishing the interval we make r tend to q; and therefore the following difference M + me x M m M+S," (7-) , dr, . . (206.) will also tend to 0, and so will also its partial differential coefficient of the first order, taken with respect to fu. We find therefore the following formula for {T, w}, (remembering that this combination has been shown to be independent of r) л 8 +m x {T} = Μq (207.) r X X r 9 M A the sign q = 9 r po implying that the limit is to be taken to which the expression tends when r tends to q. In this last formula, as in (199.), the integral 6,", dr may be considered as a known function of r, q, %, fi, or simply of r, 9,%, if be eliminated by the first condition (186.); and since it vapishes independently of x when r =q, it may be thus denoted: S,", dr = $(", 9,x) – ® (9, 9,x), x, (208.) the form of the function o depending on the law of attraction or repulsion. This integral therefore, when considered as depending on x and my by depending on x and 92 need not be varied with respect to x, in calculating {7, w} by (207.), because |