Then **qr**/ro = hk/hg or ro =W (l-x-a)/l, which is the reaction at A and shear at any point of AD, for the new position of the load.

2 **qr**., 20 cwt.

If then we put a negative point-charge -**qr**/d at B, it follows that the spherical surface will be a zero potential surface, for q rq 1 (24).

If a force Q acting at R maintains equilibrium, **QR**/4 = (P - p)r =T.

**Qr**,, be the generalized coordinates of any dynamical system, and let pi, P2,

If the system is supposed to obey the conservation of energy and to move solely under its own internal forces, the changes in the co-ordinates and momenta can be found from the Hamiltonian equations aE aE **qr** = 49 - 1 57., gr where q r denotes dg r ldt, &c., and E is the total energy expressed as a function of pi, qi,.

In the time dt which the wave takes to travel over MN the particle displacement at N changes by **QR**, and **QR**= - udt, so that **QR**/MN = - u/U.

But **QR**/MN = dy/dx.

Then the reflected ray **QR** and the ray reflected at R, and so on, will all touch the circle drawn with ON as radius.

Comparing this equation with ux 2 +vy 2 +w2 2 +22G'y2+2v'zx+2W'xy=0, we obtain as the condition for the general equation of the second degree to represent a circle :- (v+w-2u')Ia 2 = (w +u -2v')/b2 = (u+v-2w')lc2 In tangential q, r) co-ordinates the inscribed circle has for its equation(s - a)**qr**+ (s - b)rp+ (s - c) pq = o, s being equal to 1(a +b +c); an alternative form is **qr** cot zA+rp cot ZB +pq cot2C =o; Tangential the centre is ap+bq+cr = o, or sinA +q sin B+rsinC =o.

+(s - b)pq= oor - **qr** cot 2A+rptan ZB +pgtan 2C=o,with centre - ap+bq+cr = o.

If we now apply them to the case of a rigid body moving about a fixed point 0, and make Ox, Oy, Oz coincide with the principal axes of inertia at 0, we have X, u, v=Ap, Bq, Cr, whence A (B C) **qr** = L,

Since an, = a,r, the coefficient of Q, in the expression for **qr** is identical with that of Q,- in the expression for q,.

If we omit the gyrostatic terms, and write **qr** = Cre, we finc~, for a free vibration, (aj,1~2 + birX + Cm) C~ + (asrX2 + birX + Cm) C2 +

These variables represent the whole assemblage of generalized co-ordinates **qr**; they are continuous functions of the independent variables x, y, 1 whose range of variation corresponds to that of the index r, and of 1.

Equality of the angles of incidence and reflection, that the reflected ray **QR** is such that the angles RQC and CQP are equal; to determine the caustic, it is necessary to determine the envelope of this line.