This is also the general equation of virtual velocities: as to the mode of using it, observe that the displacements dp are not arbitrary quantities, but are in virtue of the mutual connexions and other geometrical conditions connected together by certain linear relations; or, what is the same thing, they are linear functions of certain independent arbitrary quantities du. Substituting for dp their expressions in terms of du we have ΣPop-Uou, where the several expressions U are each of them a linear function of the forces P, and where on the right hand refers to the several quantities du; and the resulting equation is ΣUdu = 0; viz., since the quantities du are independent, the equation divides itself into a set of equations U=0, U2 = 0,... which are the equations of equilibrium of the system.

Lagrange imagines the forces produced by means of a weight W (fig. 1) at the extremity of a string passing over a set of pullies, as shewn in the figure, viz. assuming the forces commensurable and equal to m W, nW, &c., we must have m strings at A, n strings at B, and so on. Suppose any indefinitely small displacement given to the system; each string at A is shortened by op, or the m strings at A by mop; and the like for the other particles at B, &c ; hence, if mdp+ndq+..., = (Pdp + Q&q +.....), be positive, the weight W will descend through the space

W

1

W

(P&p + Q&q +...).

Now, in order that the system may be in equilibrium, W must be in its lowest position; or, what is the same thing, if there is any displacement allowing W to descend, W will descend, causing such displacement, and the original position is not a position of equilibrium. That is, if the system be in equilibrium, the sum EPSp cannot be positive.

But it cannot be negative; since, if for any particular values of dp the sum Pop is negative, then reversing the directions of the several displacements, that is, giving to the several displacements Sp the same values with opposite signs, then the sum Pop will be positive; and we assume that it is possible thus to reverse the directions of the several displacements. Hence, if the system be in a positive of equilibrium, we cannot have ΣPop either positive or negative; that is, we obtain as the condition of equilibrium ΣP&p=0.

The above is Lagrange's reasoning, and it seems com

pletely unobjectionable. As regards the reversal of the directions of the displacements, observe that we consider such conditions as a condition that the particle shall be always in a given plane, but exclude the condition that the particle shall lie on a given plane, i.e. that it shall be at liberty to move in one direction (but not in the opposite direction) off from the plane. But the pulley-proof is equally applicable to a case of this kind. Thus, imagine a particle resting on a horizontal plane, and let z be measured vertically downwards, x and y horizontally. Suppose the particle acted on by the forces X, Y, Z, and replacing these by a weight W as above, the condition of equilibrium is, that

shall not be positive. We may have dx and dy, each positive or negative; whence the conditions X=0 and Y=0. But dz is negative; hence the required condition is satisfied if only Z is positive; that is, if the vertical force acts downwards. Clearly this is right, for if it acted upwards it would lift the particle from the plane. The case considered by Lagrange is where the particle is always in the plane; here Sz=0, and there is no condition as to the force Z.

The only omission in Lagrange's proof is, that he does not expressly consider the case of unstable equilibrium, where the weight W is at a position, not of minimum, but of maximum altitude. In such a case, however, the sum Pop is still =0, taking account (as the proof does) of the displacements considered as infinitesimals of the first order; although taking account of higher powers, the sum ΣPop would have a positive value. An explanation as to this point might properly have been added to make the proof "refutation-tight," but the proof is not really in defect.

P.S. Lagrange excludes tacitly, not expressly, the case where the direction of a displacement is not reversible; he observes that the various displacements dp, when not arbitrary, are connected only by linear equations; and "par consequent les valeurs de toutes ces quantités seront toujours telles qu'ils pourront changer de signe à la fois." The point was brought out more fully by Ostrogradsky, but I think there is no ground for the view that it was not brought out with sufficient clearness by Lagrange himself.

Parallelogram of forces.

Let P, Q, R be the forces, a, B, y their inclinations to any line; then taking ds the displacement in the direction of

this line, the displacements in the directions of the forces are Es cosa, ds cosẞ, Es cosy, and the equation EPSp=0 assumes the form

(P cosa +Q cosẞ + R cosy) ds = 0,

that is, we have

P cosa+Q cos B+ R cosy = 0,

viz. this equation holds whatever be the fixed line to which the forces are referred. It is easy to see that, supposing it to hold in regard to any two lines, it will hold generally, and that the relation in question is thus equivalent to two independent conditions; and forming these we may obtain from them the theorem of the parallelogram of forces.

But to obtain this more directly, take A, B, C for the angles between the forces Q and R, R and P, P and Q respectively, then A+B+C=2π, and thence

a = α,

B=a+C,

y=a+C+A= a + 2π- В,

whence writing a=π, or taking the line of displacement at right angles to the force P, we have

a=1π, B={π+ С, y=2π+žπ – B,

and the equation becomes OP-Q sin C+ R sin B=0, that is QR sin B: sin C; and similarly R: P= sin C: sin A, that is

=

P: Q; R = sin A: sin B: sin C,

equations which in fact express that each force is equal and opposite to the diagonal of the parallelogram formed by the other two forces.

MATHEMATICAL NOTES.

8

Geometrical proof that 13 + 23 +33 + n3 = (1 + 2 + 3 ... + n)3. The above proposition follows very simply from a geometrical diagram as follows. It will be shown that by increasing the side of the square whose side is 1+ 2... + (n − 1) by n, the area of the square itself is increased by n squares each equal to a square on side n. In (fig. 2), the mode of formation of the successive squares in which is self-evident, suppose ABCD is the square on 1+2+... (n−1); LB being n-1, and suppose further that the gnomon LCM is equal to n-1 squares, each on side n-1; then it obviously follows that the rectangle CL is equal to +1(={n) squares, each on side n n-1 2

n-1

2

1, so that on CB there would be room for

the sides of squares, of each on side n.

Take BK=n and complete the square, whence we have the rectangle CK

equal to

n 1

2

squares each on side

of the gnomon BHE is equal to 2.

senting the square FCHG in the corner) viz. to n squares, each on side n. The proposition thus follows by induction, since all the suppositions are evidently realised when n=2 or 3. It will be noticed that the dotted lines in the figure divide the areas that form the differences between the successive squares into n complete, or n-1 complete and two half squares, according as n is odd or even.

is evident by arranging points in the form of a triangle with 1, 2, 3 ... in successive rows; but it also is seen by forming a rectangle, as in (fig. 3), in which the succeeding parts contain 2, 4, 6 ... equal squares, the area of the whole rectangle being n (n+1); whence the result.

J. W. L. GLAISHER.

Relations between the angles of regular bodies.

By aid of Vega's ten-figure logarithmic tables, I found, in 1863,

By combining these and other Euclidean angles I have produced some wonderfully close approximations. Ex. gr.

§ (3T+ D + I ) − 113° 15′ = 57° 17′ 44′′.759167,

or arc = radius to a four-millionth, as 1131 = 90 +114 + 12, 90° (2D+31-3T+15°) = 51° 25′ 43′′.053644, or heptagon angle + † (1′′.375508) true to 1-942000,

(10T+91− 3D + 22° 30′) + 72° = 124° 48′ 21′′.797443, exceeding arc (chord. sq. root of π) by 1".097 only.

=

These and others can be marked by simple repetition on a circle, where radius is bisected and trisected by perpendicularly cutting the circumference.

Logarithmic and factor tables.

S. M. DRACH.

I strongly suggest the paramount utility of publishing, as a companion octavo volume to Hutton and other logarithmic tables, the sixth to the tenth decimals of Vega's large folio volume; this rare work is only found in the Museums of Capital Cities, instead of a ten shilling book in the home of every student that requires them.

I would improve the tripartite division of factors in Vega, Burckhardt, etc., by this continuous one of only eight vertical columns.

showing the sequences of 7, 11, 13, etc. without dislocation.

74, Offord Road, N. (London),

31 March, 1873.

S. M. DRACH.