the same functions of r, s respectively, and m and n being integers (or more generally for other relations between the forms of R, S given by the theory of elliptic integrals), the equation admits of algebraical integration; but as the relations in question do not in general hold good, the theory of the algebraical integration of the equations plays only a secondary part in the solution of the problem. It is, however, proper to remark that Euler, when he wrote his first two memoirs "On the Problem of the two Centres" (post, Nos. 45 and 46), had already discovered and was acquainted with the theory mdr nds WR S

of the algebraic integration of the equation =

(R, S, m, n, ut suprà),

dy
√A+By+Cy2+Dy3+Ey1

N. Comm. Petrop. t. xii. 1766-1767?, bears in fact a somewhat later date. 45. Having made these preliminary remarks, I come to the history of the problem.

It is I think clear that Euler's earliest memoir is the one "De Motu Corporis &c." in the Petersburg Memoirs for 1764 (printed 1766). In this memoir the forces vary as (dist.)-2, and the body moves in a given plane. The equations of motion are taken to be

which, if ", ŋ are the inclinations of the distances v, u to the axis respectively (see foot-note to No. 42), lead to

v2u2 dl dn=2gadt (A cos +B cos n+D),

where D, E are constants of integration. Substituting for v, u their values in terms of ŋ, and eliminating dt, Euler obtains

where

desin n P+/ P2 — Q2
dn sin
Q

A cos n+B cos +D cos cos n+E sin & sin n=P,

A cos +B cos n+D

=Q.

And he then enters into a very interesting discussion of the particular case A=0 or B=0 (viz. the case where one of the attracting masses vanishes, which was of course known to be integrable); and after arriving at some paradoxical conclusions which he does not completely explain, although he remarks that the explanation depends on the circumstance that the integral found is a singular solution of a derivative equation, and as such does not satisfy the original equations of motion,-he proceeds to notice that an inquiry into the cause of the difficulty led him to a substitution by which the variables were separated.

46. But in the memoir "Problème, un Corps &c." in the Berlin Memoirs for 1760 (printed 1767), after obtaining the last-mentioned formula, he gives

at once, without explaining how he was led to it, the analytical investigation of the substitution in question, viz. in each of the two memoirs he in fact writes

p=tantan; q=tan÷tan n;

and in terms of these quantities p, q, the equation becomes

P=(A+B+D)p+2Ep2+(-A−B+D)p3,

Q=(-A+B-D)q+2Eq2+( A-B-D)q',

so that P and Q are cubic functions (not the same functions) of Р and q respectively; and the equation for the time is found to be

which are the formulæ for the solution of the problem, as obtained in Euler's first and second memoirs.

47. In his third memoir, viz. that " De Motu Corporis &c." in the Petersburg Memoirs for 1765 (printed 1767), Euler considers the body as moving in space, the forces being as before as (dist.)-2. Assuming that the coordinates y, z are in the plane perpendicular to the axis, there is in this case dz dy dt

the equation of areas y t -z =const. ; and writing y=y'sin y, z=y' cos,

=

that is, y'y2+2, and the azimuth, the integral equations for the motion in the variable plane (coordinates a, y) are not materially different in form from those which belong to the motion in a fixed plane, coordinates x, y (see post, No. 56, Jacobi); and the last-mentioned equation, which reduces

αψ
dt

itself to the form y' =const., gives at once d in a form such as that

by quadratures.

above alluded to (antè, No. 43), and therefore
variables employed by Euler in the memoir in question are

vu, v-u (say r, s) and ↓,

The

v, u being, as above, the distances from the two centres, and the azimuth of the axial plane. The functions of r,s under the radical signs are of the fourth order; this is so, with these variables, even if the motion is in a fixed plane; but this is no disadvantage, since, as is well known, the case of a quartic radical is not really more complicated than that of a cubic radical, the two forms being immediately convertible the one into the other. 48. Lagrange's first memoir (Turin Memoirs, 1766-1769) refers to Euler's three memoirs, but the author mentions that it was composed in 1767 without the knowledge of Euler's third memoir. The coordinates ultimately made use of are v+u, v-u (say r, s) and 4, the same as in Euler's third memoir, and the results consequently present themselves in the like form.

49. If the attractive force of one of the centres is taken cqual to zero, then the position of such centre is arbitrary, and it may be assumed that the centre lies on the curve, which is in this case an ellipse (conic section); the expression of the time presents itself as a function of the focal radius vectors and the chord of the arc described; which, as remarked, antè, No. 20, leads to Lambert's theorem for elliptic motion.

50. The case presents itself of an ellipse or hyperbola described under the dr ds = will be satisfied WR S

action of the two forces, viz. the equation

by r—a=0, if r-a be a double factor of R, or by s-ß=0, if s—ß be a double factor of S, a case which is also considered in the Mécanique Analytique;' and see in regard to the analytical theory, t. ii. 3rd ed. Note III. by M. Serret, and "Thèse," Liouv. 1848. It is remarked by M. Bonnet, Note IV. and Liouv. t. ix. p. 113, 1844, that the result is a mere corollary of a general theorem, which is in effect as follows, viz. if a particle under the separate actions of the forces F, F', . . . starting in each case from the same point in the same direction but with the initial velocities v, v', &c. respectively, describe the same curve, then such curve will also be described under the conjoint action of all the forces, provided the body start from the same point in the same direction, with the initial velocity V=q/v2+v”+••

51. Lagrange's second memoir (same volume of the Turin Memoirs) contains an exceedingly interesting discussion as to the laws of force for which the problem can be solved. Writing U, V, u, v in the place of Lagrange's P, Q, p, q, the equations of motion are

and putting also ƒ (= √(a−a)2+(b−ß)2+(c—y)2) the distance of the centres,

U

V

ν

and then u2=ƒ3x, v2=ƒ3y, z=X, z=Y (x,y are of course not to be confounded with the coordinates originally so represented), Lagrange obtains the equations

and he then inquires as to the conditions of integrability of these equations, for which purpose he assumes that the equations multiplied by mdx+ndy and uda+rdy respectively and added, give an integrable equation. 52. A case satisfying the required conditions is found to be

2au, 2av, varying directly as the distance, and of the same amount at equal distances; or, what is the same thing, there is, besides the forces varying as (dist.)-2, a force varying directly as the distance, tending to a third centre midway between the other two, a case which is specially considered in the memoir; it is found that the functions in r, s under the radicals (instead of rising only to the order 4) rise in this case to the order 6. 53. Among other cases are found the following, viz. :—

In regard to the subject of this second memoir of Lagrange, see post, Miscellaneous Problems, Liouville's Memoirs, Nos. 100 to 105.

54. In the Mécanique Analytique' (1st ed. 1788, and 2nd ed. t. ii. 1813), Lagrange in effect reproduces his solution for the above-mentioned law of 6

α

2

force (say U= +2yu, V=+2y)*. There are even in the third edition a few trifling errors of work to be corrected.

The remarks above referred to, as made by Lagrange in his first memoir, are also reproduced (see antè, Nos. 49 and 50).

55. Legendre, "Exercices de Calcul Intégral," t. ii. (1817), and "Théorie des Fonctions Elliptiques," t. i. (1825), uses p2 and q2 in the place of Euler's p, q; the forces are assumed to vary as (dist.)-2, and in consequence of the change Euler's cubic radicals are replaced by quartic radicals involving only even powers of p and q respectively; that is, the radicals are in a form adapted for the transformation to elliptic integrals; in certain cases, however, it becomes necessary to attribute to Legendre's variables p and q imaginary values. The various cases of the motion are elaborately discussed by means of the elliptic integrals; in particular Legendre notices certain cases in which the

* In the Mécanique Analytique,' Lagrange's letters are r, q for the distances r+q=8, r-q=u: the change in the present Report was occasioned by the retention of p, q or Euler's variables.

motion is oscillatory, and which, as he remarks, seem to furnish the first instance of the description by a free particle of only a finite portion of the curve which is analytically the orbit of the particle; there is, however, nothing surprising in this kind of motion, although its existence might easily not have been anticipated.

56. § 26 of Jacobi's memoir "Theoria Novi Multiplicatoris &c." (1845) is entitled "Motus puncti versus duo centra secundum legem Neutonianum attracti." The equations for the motion in space are by a general theorem given in the memoir " De Motu puncti singularis" (1842), reduced to the case of motion in a plane: viz. if x, y are the coordinates, the centre point of the axis being the origin, and y being at right angles to the axis, and if the distance

day

of the centres is 2a; then the only difference is that to the expression for dt2 a2 there is added a term which arises from the rotation about the axis. Two ya, integrals are obtained, one the integral of Vis Viva, and the other of them an integral similar to one of those of Euler's or Lagrange's. And then x', y' being the differential coefficients of x, y with regard to the time, the remaining equation may be taken to be y'dx-x'dy=0, where x', y' are to be expressed as functions of x, y by means of the two given integrals. This being so, the principle of the Ultimate Multiplier furnishes a multiplier of this differential equation, and the integral is found to be

*

2 — y'') + (a2 — x2+y2) x'y'

€,

the quantity under the integral sign being a complete differential. To verify à posteriori that this is so, Jacobi introduces the auxiliary quantities X', X' defined as the roots of the equation λ2+λ(x2+y2—a2)—a2y2=0, which in fact, if as before u, v are the distances from the centres, leads to

so that X, X" are functions of u+v, u—v respectively; and the formulæ, as ultimately expressed in terms of X', A", are substantially of the same form with those of Euler and Lagrange.

57. The investigations contained in Liouville's three memoirs "Sur quelques cas particuliers &c." (1846), find their chief application in the problem of two centres, and by leading in the most direct and natural manner to the general law of force for which the integration is possible, they not only give some important extension of the problem, but they in fact exhibit the problem itself and the preceding solutions of it in their true light. But as they do not relate to this problem exclusively, it will be convenient to consider them separately under the head Miscellaneous Problems.

58. In Serret's Thèse sur le Mouvement &c.' (1848), the problem is very elegantly worked out according to the principles of Liouville's memoirs as follows: viz. assuming that the expression of the distance between two consecutive positions of the body is

ds2=λ(mdμ2+ndv2)+λ"dy2,

where m, n are functions of μ, v respectively, and if the forces can be represented by means of a force-function U, then the motion can be determined,

Explained in Jacobi's memoir "Theoria Novi Multiplicatoris &c.," Crelle, tt. xxvii. xxviii. xxix. 1844-45.