point is the same as it would be if the two forces were transferred to the point, retaining their direction and magnitude.

Let P, Q, be two forces, acting to turn a body about a fixed point C. Draw CA

parallel to the force P and CB parallel to the force Q; the pressure on C is the same as if the forces P, Q, acted in the lines AC, BC.

Produce the directions of the forces to meet in D, and complete the parallelogram CADB.

B

D

The force P, produces the same effect as if it acted at the point D in P's direction by Axiom 8; and similarly the force Q produces the same effect as if it acted at D. And if Dp, Dq represent the forces P, Q, and the parallelogram Dprq be completed, the diagonal Dr will represent the force at D to which P and Q are equivalent. But the direction of the force Dr must pass through the point C, as in Prop. 12, and will produce the same effect as if it acted at. C; and the force Dr acting at C is equivalent to the forces qr, pr, acting in directions parallel to qr, pr, by Prop. 13; that is, the force Dr is equivalent to the forces Dp, Dq, acting in the lines AC, BC; that is, the forces P, Q, acting in the lines BP, AQ are equivalent to forces P, Q_acting in AC, BC. Therefore the pressure upon the fixed point C is the same as if the forces P, Q were transferred to that point. Q. E.D.

COR. 1. If, instead of the fixed point at C, we substitute the pressure which that point exerts, there will be equilibrium by Axiom 12. Hence, if a body be acted upon by three forces in the same plane, of which one passes through the intersection of the other two,

and is equal to the resultant of the other two, the body will be in equilibrium.

COR. 2.

Conversely if there be equilibrium, these conditions obtain. This follows from Axiom 3.

PROP. XXXIV. If two forces tending to turn a body round a fixed axis, and acting in two planes perpendicular to the axis, balance each other, (as in the Wheel and Axle,) the pressures upon the points of the axis where the body is supported, are the same as they would be, if the two forces, retaining their direction and magnitude, were transferred to the axis, at the points where the perpendicular planes meet it.

Let P, Q, be two forces acting perpendicularly

same as if the forces P and Q, continuing parallel to themselves, were transferred to C and D.

Let AC be produced to F, CF being equal to CA, and at F in the plane PAC, and perpendicular to DG, let two forces P', P', each equal to P, act in opposite directions. These forces will balance each other and be equivalent to no force; and therefore if the forces

P', P' are added to the system, the equilibrium will not be disturbed.

In like manner produce ED to G, DG being equal to ED, and at G, in the plane QED, and perpendicular to DG, let two forces Q', Q", each equal to Q, act in opposite directions: these forces will not disturb the equilibrium. Therefore the six forces P, P', P, Q, Q', Q", acting in the manner described, will be supported by the forces X, Y; that is, the eight forces P, P', P'', Q, Q', Q", X, Y, balance each other.

The forces P", Q", are situated in exactly the same manner with regard to vertical lines and planes drawn upwards, as P, Q are, with regard to vertical lines and planes drawn downwards. Therefore P", Q", would balance each other on the axis HK, and would produce at H and K pressures equal and opposite to those which P, Q produce. But the forces X, Y are equal and opposite to the pressures which P, Q produce, for they balance those pressures. Therefore the forces P", Q" produce at H, K the pressures X, Y.

The forces P, P' are equivalent to a force double of P acting at C, parallel to P; and the forces Q, Q' are equivalent to a force at D double of Q, parallel to Q.

Hence the six forces P, P', P', Q, Q', Q" are equivalent to X, Y, at H, K, and to 2P, 2Q at C, D. And the eight forces P, P, P, Q, Q', Q", X, Y are equivalent to 2X, 2Y at H, K, and to 2P, 2Q, at C, D.

But these eight forces balance each other; therefore 2X, 2 Y, acting at H, K, balance 2 P, 2 Q, acting at C, D: and therefore X, Y, which balance P, Q, acting at A, E, would balance P, Q, acting at C, D. Q. E. D.

72

1.

BOOK II. HYDROSTATICS.

DEFINITIONS AND FUNDAMENTAL NOTIONS.

HYDROSTATICS is the science which treats of

the laws of equilibrium and pressure of fluids.

* 2. Fluids are bodies the parts of which are moveable amongst each other by very small forces, and which when pressed in one part transmit the pressure to another part.

*3. Some fluids are compressible and elastic; that is, they are capable of being made to occupy a smaller space by pressure applied to the boundary within which they are contained, and when thus compressed, they resist the compressing forces and exert an effort to expand themselves into a larger space. Air is such a fluid.

* 4. Other fluids are incompressible and inelastic; not admitting of being pressed into a smaller space nor exerting any force to occupy a larger. Water is considered as such a fluid in most hydrostatical reasonings.

5. In all fluids which have weight, the weight of the whole is composed of the sum of the weights of all the parts.

AXIOMS.

1. If a fluid of which the parts have no weight be contained in a tube of which the two ends are similar and equal planes, two equal pressures applied perpendicularly at the two ends will balance each other.

Let ABCD be the tube, AB, CD its two equal ends:

the equal forces P, Q, acting

B

perpendicularly on these ends will balance each other.

2. If a fluid be at rest in any vessel, and if any forces, acting on two portions of the boundary of the fluid, balance each other, they will also balance each other if any portions of the fluid become rigid without altering the magnitude, position, or weight of any of their parts.

Thus if the two forces P, Q, acting on AB, CD, parts of the surface p of a vessel containing

fluid, balance each

other; they will also

balance each other if

A

B

D

C

the parts E and F of the fluid be supposed to become rigid, the magnitude, position and weight, of all the parts of E, F, remaining unaltered.

3. If two forces acting upon two portions of the boundary of a fluid balance each other, and if a force be added to one of them, it will prevail and drive out the fluid at the part of the surface acted on by the other force.

4. Any surface pressed by a fluid may be divided into any number of particles, and the pressure on the whole is equal to the sum of the pressures on each of the particles.

5. When a plane surface is pressed by a fluid, the pressure exerted on the surface, and the pressure of the surface on the fluid are perpendicular to the plane.

6. We may reason concerning fluids, supposing them to be without weight: and we shall obtain the pressures which exist in heavy fluids, if we add, to the pressures which would take place if the fluids had no weight, the pressures which arise from the weight.

D