A C 2BC 2 And from which we deduce, that so long as the velo city is not sufficient to produce an elevation of the water before, and a depression abaft the greatest breadth, so as to increase the fore resistance and di minish the aft, the body will experience the same re sistance, whether the sharp or obtuse extremity moves forward; and yet that the resistance will be the least when the two extremities are equal, or when the greatest breadth is in the middle.
But. if we suppose that the water runs ahead of the ship before its greatest breadth, with a velocity repre sented by e, and that it has acquired a velocity iv in a direction opposite to that of the body abaft this greatest breadth, then the velocity forward will be denoted by n — v, and the velocity aft, by n w.
Since, moreover, the resistance is in proportion to the squares of the velocities,t the general expression for the resistance will be changed to 2 D2 (n v)2 C(n w)2 2 AC where we suppose C' to be greater than C, inasmuch as the water before the greatest breadth is more ele vated than behind it.
Hence it is evident, that whatever relation exists between the elements 71, v, and zv, the body meets with less resistance when the obtuse end is forward, than when the acute end is so placed; and that the position of the main breadth with respect to the middle point of the body, depends on the quantities 71— V and in order that the resistance may be less than if its situation were anywhere else.
It is manifest also, that the greater v and tv arc with respect to n, the more the greatest breadth should be carried before the middle, in order to render the re sistance the least.
No supposition can be imagined in which the ele ments n and v become identical; because in such a case, the water would flow forward with an equal ve locity to the ship, a condition by no means admissible. It is also to be remarked, that a becomes very small with respect to n, when the velocity is small, and may even be supposed to vanish when n becomes very mi nute. The same thing also takes place with respect to the water abaft the greatest breadth, when the ve locity is small, and the body has its maximum breadth very far aft, the water following the body to fill up the void space which it leaves. this cause a
part of the water follows the same direction as the body, the velocity of the body in relation to the water being n — w, and which therefore gives to the pre ceding expression for the resistance, the still more general form of Such is the general expression for the resistance as given by Chapman,—an expression which it is pro per to apprise our readers cannot be entirely depend ed on, inasmuch as he admits in it two suppositions, which have been repeatedly proved by experiment to be false. The first of' these suppositions is, that the resistance to a plane surface varies as the square of the sine of the angle, which the surface makes with the line of motion; and secondly, when the angle is given that the resistance varies as the square of the velocity.
This general expression for the resistance is subject to different varieties of value, according to the rela tive dimensions of AC and BC; and there is moreover involved in its consideration, the position of the great est breadth, when the resistance itself is a minimum, —a principle of great importance in the practice of shipbuilding. Dr. Inman, by reducing the expres sion for the resistance to a differential equation, has found, when the length is four times the breadth, that the greatest transverse section of a ship ought to be about of the length before the middle; or when the length is to the breadth as 3 to 1, about of the length before the same point.
Chapman deduces the conclusion, that to render the resistance always a minimum, the place of' the greatest breadth ought to vary,—a condition obvious ly impossible.
In order, however, to put our readers in possession of the method employed by Chapman for estimating the resistance on a ship, we shall adopt his co-effi cients of resistance, that for the forepart of the vessel being 6, and for the afterpart 7.* The expression for the resistance on the forepart. of the vessel will there fore be Dcz 6C x--- and on the after-part 7 C x -----. These expressions we shall now endeavour to apply.