Resistance to the cosine of the angle K T n = ; (radius being unity a cylinder moving obliquely. in all cases). Now let z = FT and 2 the fluxion of the same, then the fluxion of the force of the fluid on FT will be = n v 2 s 2 ƒ 1 ż multiplied by the sine of the angle PTn, whereof the angle PTn being composed of the two angles P TQ, QTn, the natural sines and cosines of which are represented above; its sine by trig. will be expressed by Therefore which corrected will, in the ultimate case, where x = r, be which is therefore the effective force of the fluid on the quadran force on the whole semicylindric surface m Dur Bs; OT fr h 6 g (c + 2) the or the resistance to the cylinder when moving in the fluid Resistance to at rest, so far as relates to that surface. a cylinder moving ob To determine what farther resistance is opposed to the liquely. cylinder by the fluid acting against the top As Br. Let us suppose AV BT (fig. 3) to be the head of the cylinder, and a particle striking it at T; also let A B be a diameter of the circle perp. to the axis, and draw T Q parallel to A B, and PQ and QR perp. to TQ and TP respectively. Then PT being considered the representative of the full force of a particle, and to be resolved into the two forces P Q, TQ; the force TQ, being parallel to the plane A B V, has no effect in causing it to move; but only the force denoted by PQ, which is as the sine (c) of the angle P TQ.. Therefore the effective force of a particle in this case will be ; and that of the fluid on the whole circular plane 4g nv2 c3 pr1 4g 1 (p being = 3.1416). Hence the whole resist Cor. 1. When the angle T P Q (fig. 2) is 90°, or the solid moves in a direction perp. to its axis; then becoming 1 and c nothing, the resistance to the cylinder will be nv2r h 3g as determined in the first lemma. Cor. 2. The resistance to the cylinder moving in the di rection T P estimated in the direction QT is 6g (e+2), being that arising only from the action of the fluid upon the semisurface of the solid; that on the head or top of the cylinder having no effect to move it inthis direction, but in the direction of its axis. For an example to this proposition in numbers, when the medium is supposed to be that of our atmosphere. Let the angle TPQ (the sine of which is Л) = 60°; and consequently the angle PTQ (the sine of which is c) = 30°. Then 03196687 + 00187486 ≈ 03384173 ounces for the resistance to a cylinder of the above dimensions, when moving with the velocity of 1 foot per second. And therefore, as the resistance to the same cylinder varies as the square of the velocity, the resistance corresponding to any other velocity will be had by multiplying the above by the velocity (in feet) squared. A quadratic equation ap three roots. On the Defective Algorithm of Imaginary Quantities. In a SIR, To Mr. NICHOLSON. IN a mathematical investigation, in which I was lately engaged, I fell upon a very singular anomaly in the theory of equations, which is nothing less than a quadratic equation having (at least to all appearance) three roots, all different from each other; whereas, according to received principles, it can have only two. As this is a very ftrange deviation from what has been hitherto considered as a well established, theory, I am induced to request the publication of it in your Journal, in hopes that some of your mathematical correspondents may undertake to explain the difficulty, and rescue the theory of equations, and the present algorithm of imaginary quantities, from the danger to which such anomalies muft necessarily expose them; particularly as there are are some among us, who wish to cramp the power of analysis, by rejecting in that science every species of quantity coming under an imaginary form. I think I can perceive where the mystery lies; but still I should be glad to see the opinion of more able analysts on this apparent incongruity; if however no such should appear, I will, through the medium of your Journal, publish my ideas on the subject. The equation to which I have alluded is this: The two first of which evidently answer the conditions of the equation, and with the third I proceed as follows. And now in order that I may be certain of my results, I multiply these quantities under the radicals at full length, as follows; viz. ++✓ for the product (+4) × (t−√ ̄ ̄?) Now if this be a legitimate result, I see no reason why this value of r should not be considered as a root of the proposed equation as well as the other two; and if it be admitted as such, then I can find any number of other roots at pleasure; which will totally destroy the established theory of equations; but if, on the contrary, this cannot be admitted as a root, then it necessarily follows, that the present algorithm of imaginary quantities is defective, or otherwise that I have deviated from that algorithm in the preceding operation. In order to discover the errour wherever it may lie, and that the connection of it may be made public, I am induced to request the publication of this paper in your Journal; which, if you should think proper to comply with, will much oblige Yours &c. MATHEMATICUS. |