two spherical surfaces intersect; the two surfaces having equal radii.

=

If a the semi-axis of the lens, b = the radius of the circular intersection of the two surfaces; k the radius of gyration of the lens about its axis, and k' about a diameter of the circle; we shall have

SECT. 8. Principal Axes of a Plane Lamina at any
proposed point in the Lamina.

One of the axes is at right angles to the lamina. Let x, y, be the co-ordinates of any infinitesimal element m of the lamina, referred to axes in the plane of the lamina which pass through the point. Then, denoting the inclination of either of the other two principal axes at the point to the axis of x,

(1) To find the principal axes of the area of a lamina, in the form of a semi-loop of a lemniscate, at the node.

Thus the two principal axes, which lie in the plane of the lamina, are inclined to the axis of the lemniscate at angles

(2) To find the principal axes at an end of a wire in the form of a semicircle.

One of the axes is at right angles to the plane of the wire, and the inclinations of the other two to the chord of the semicircle are given by the equation

(3) To find the principal axes of a right-angled triangle at the right angle.

One of the axes is perpendicular to its plane and the other two are in its plane and are inclined to its sides at angles equal where a is one of the acute angles of the

1 -1 tanTM1

to

2

triangle.

(tan 22),

Griffin; Solutions of the examples on the motion of

a Rigid Body, p. 8.

(4) A parabolic area is included between the curve, the axis, and the semi-latus rectum: to find the positions of the principal axes of the area at the vertex of the parabola.

If be the inclination of either principal axis, in the plane of the area, to the axis of the parabola,

(5) To find the principal axes at a point in the circumference of an elliptic lamina.

One of the axes is at right angles to the lamina and, denoting the inclination of either of the other two to the major axis,

where x and y are the co-ordinates of the point referred to the axes of the ellipse as axes of co-ordinates.

Griffin; Solutions of the Examples on the motion of a Rigid Body, p. 8.

CHAPTER VI.

D'ALEMBERT'S PRINCIPLE.

A GENERAL method for the determination of the motion of a material system, acted on by any forces, was laid down by D'Alembert in his Traité de Dynamique, published in the year 1743', from which we have extracted the following passage in exposition of the Principle.

"Problême Général.

"Soit donné un systême de corps disposés les uns par rapport aux autres d'une manière quelconque; et supposons qu'on imprime à chacun de ces corps un mouvement particulier, qu'il ne puisse suivre à cause de l'action des autres corps; trouver le mouvement que chaque corps doit prendre.

"Solution.

"Soient A, B, C, &c. les corps qui composent le systême, et supposons qu'on leur ait imprimé les mouvemens a, b, c, etc. qu'ils soient forcés, à cause de leur action mutuelle, de changer dans les mouvemens a, b, c, etc. Il est clair qu'on peut regarder le mouvement a imprimé au corps A comme composé du mouvement a, qu'il a pris, et d'un autre mouvement a; qu'on peut de même regarder les mouvemens b, c, etc. comme composés des mouvemens b, B; c, x; etc. d'où il s'ensuit que le mouvement des corps A, B, C, etc. entr'eux auroit été le même, si au lieu de leur donner les impulsions a, b, c, etc. on leur eût donné à-la-fois les doubles impulsions a, a; b, B; c, x, etc. Or par la supposition, les corps A, B, C, etc. ont pris d'eux-mêmes les mouvemens a, b, c, etc. donc les mouvemens α, ß, x, etc. doivent être 1 See also his Recherches sur la Précession des Equinoxes, p. 35, published in 1749.

2 D'Alembert's Principle was first enunciated by him in a memoir which he read before the Academy of Sciences at the end of the year 1742.

tels qu'ils ne derangent rien dans les mouvemens a, b, c, etc. c'est-à-dire que, si les corps n'avoient reçu que les mouvemens a, ẞ, κ, etc. ces mouvemens auroient dû se détruire mutuellement, et le systême demeurer en repos.

"De là résulte le principe suivant, pour trouver le mouvement de plusieurs corps qui agissent les uns sur les autres. Décomposez les mouvemens a, b, c, etc. imprimés à chaque corps, chacun en deux autres a, a; b, B; c, K; etc. qui soient tels, que si l'on n'eût imprimé aux corps que les mouvemens a, b, c, etc. ils eussent pu conserver ces mouvemens sans se nuire réciproquement; que si on ne leur eût imprimé que les mouvemens a, ß, ê, etc. le systéme fût demeuré en repos; il est clair que a, b, c, etc. seront les mouvemens que ces corps prendront en vertu de leur action. Ce qu'il falloit trouver."

et

The idea of the general method developed by D'Alembert for the determination of the motion of material systems, had occurred somewhat earlier to Fontaine, as we are informed in the Table des Mémoires, prefixed to his Traité de Calcul Différentiel et Integral', having been communicated by him to the Academy of Sciences in the year 1739, and subsequently to several mathematicians. His views, however, on this subject were not made public till long after the appearance of the Traité de Dynamique; and in all probability D'Alembert, who did not become a member of the Academy before the year 1741, was not aware of Fontaine's generalization. D'Alembert, however, was the first to shew the wonderful fertility of the Principle by applying it to the solution of a great variety of difficult problems, among which may be mentioned that of the Precession of the Equinoxes, which had been inadequately attempted by Newton, and of which D'Alembert was the first to obtain a complete solution.

The earliest step towards the discovery of D'Alembert's Principle is to be met with in a memoir by James Bernoulli in the Acta Eruditorum, 1686, Jul. p. 356, entitled "Narratio Controversiæ inter Dn. Hugenium et Abbatem Catelanum agitatæ de Centro Oscillationis quæ loco animadversionis esse poterit in

1 Mémoires de l'Académie des Sciences de Paris, 1770.