The last column denominated the multiplier, is a number which AIarestier deduced, to show the relation which the true velocity of a boat bears to the follow ing quantity: The square root of the product of the height of the column of mercury the steam will sup port, the stroke of the piston, and the square of its diameter, divided by the square root of the product of the rectangle of the breadth and draught of water of the vessel, and the diameter of the paddle wheel.
We shall now endeavour to explain the principles by which Al. AIarestier deduces so ingeniously the terms factor and multiplier, the values of which arc recorded in the two last columns of the preceding table.
To accomplish this he supposes the motion of the vessel to become uniform, and the force of the steam con stant; and on this hypothesis, and the data he has col lected in the preceding- table, lie investigates the pro portions which exist between the power of the engine, the dimensions of the vessel, of the paddles and the wheel. Ile assumes, moreover, that the resistance of the paddles is equal to the resistance of a surface moved in the Add in a direction popendicalar to itself, and hav iag a velocity erred to the mean rcheity of the paddle. This surface, which he denominates the resisting sur face of the paddles, is represented by , . The velocity of the resisting surface by U The resisting surface of the vessel by 62 And the velocity of the vessel by Each of these quantities he proposes to derive from experiment.
I. The resistance of the hull being supposed pro portional to the square of the velocity, is equivalent to k the function k being the measure of the direct resistance corresponding to the unity of surface and velocity.
Then the velocity with which the paddles strike the fluid being the resistance they experience will be k (u_v)2.
Hence it follows, that k = k (1.: , The velocity of the vessel is therefore always pro portional to that of the paddles, while the resisting surface of the vessel bears a constant relation to the surface of the paddles.
2. The moments arising from the action of the paddles on the water, and the steam on the piston, are equivalent to each other, omitting the effects of fric tion. The absolute velocity of the paddles being also U, and the resistance they meet with k (U the moment of their action, will be k (U_ U.
Supposing q to represent the density of the mercury, h the altitude of the column the steam will support, P the surface of the piston, and v the measure of its mean velocity; then will the moment of the piston be equi valent to q h P and consequently q h P v =k (U_ U.
3. Since the effect of the friction of the machine is to diminish the effect of the moving force communi cated from the piston to the paddles, a portion only of the moving force q h P is taken, and which is re presented by 2/2 q h P. Hence we obtain the equation mq1iPv=k (U U.
and since U V, a we obtain by the necessary reductions, V 3 I( inglzPv kb2(-4-)' and a b ) 3 h P v 1 b 2 cn 4. From these formulae we may draw the following the cube of the velocity of the vessel is less than the power of the engine, divided by the re sistance of the vessel; and that the cube of the mean velocity of the paddles is also greater than the same limit only to be attained when the paddles are infinite.
5. If we suppose a second boat to exist, the ele ments U', V' a', b', Sze. of which are analogous to
those of V, a, b, kc. adopted for the former boat, we may obtain by the common processes of reduction b - V. a 3 v zn h P o b' a' = 4 2,1 3h p v ' v' b 2 b')) ± a' and a So also when the resisting surfaces of the paddles are, in both vessels, proportional to the resisting surfaces of their hulls, ' we obtain b = a' a 'V' U' P' v' and consequently =-- = / ( _ V U 1/ (va h P v 112 / Hence it follows, that the velocities of the boats are proportional to the velocities of the paddles, (kid they are also in a direct proportion to the cube root of the pow& of the engines, and in an inverse proportion to the cube root of the resistance of the vessels. Al. Alarestier considers this proposition nearly general; because, un less there is a very great disproportion in the dimen b' sion of the vessels, the relation of 1+ a a'not differ much from unity.
Throughout these investigations, Al. Alaresticr has regarded h as the altitude of the column of mercury, which the steam when acting on the piston will sup port, and determined the effort of the piston, under the supposition that the vacuum on the contrary side of the piston is perfect; but as such a condition can not exist, the quantity h should be diminished by the height which the steam remaining on the contrary side of the piston, will depress the mercury from the altitude at which it would stand in a common barome ter. This is an important consideration when com paring one boat With another, because the degree of the vacuum must depend wholly on the goodness of the engine.
6. From the equations b V = a (U V), and mqhPv=k0 (U mq/iPv we may deduce UV`-' = Therefore whatever may be the dimensions of the paddles, the product of their velocity and the square of the velocity of the vessel is in proportion to the power of the engine.
Although the power of the engine has been consi dered as known, it is seldom that the velocity of the piston can be taken arbitrarily. The relation of this velocity to that of the paddles is almost always in variable, and therefore the velocity of the piston al ters with any increase or diminution in the size of the paddles. This however will not make any change in the conditions of the preceding question; but the value of v will vary according to the alteration. It may happen either that the velocity of the piston is too great to admit of an adequate supply of steam, or that the supply of vapour is too great, and some ne cessarily escapes by the safety valve. In the first case, the elastic force of the vapour will diminish until the movement of the piston shall correspond to the quan tity of steam supplied; and in the second case to pre vent the loss of steam, the intensity of the fire must be diminished; but then the power of the engine will be reduced in the proportion of the actual velocity of the piston to that which it ought to have.
That the velocity of the piston may correspond to the quantity of steam furnished by the boilers, the mechanism must be so arranged as to satisfy the equation 3 mqhPo b 2 U = 4( k 1 + c7) ) or if r represents the relation between the velocities of the piston and paddles, we may obtain the equation 3 7/2 q h P 2r = = vT( a Of the quantities a, b, It, I', r, U, V, and r contained in the equations.
U = -I- /) l V, a