276 THE MATHEMATICAL PRINCIPLES [BOOK II. one case, are found, the hyperbola may afterwards be easily determined from any given angle NAH. RULE 2. If the angle NAH, and the density of the medium at A, re main the same, and the velocity with which the body is projected be changed, the length AH will continue the same ; and AI will be changed in a duplicate ratio of the velocity reciprocally. RULE 3. If the angle NAH, the velocity of the body at A, and the accelerative gravity remain the same, and the proportion of the resistance at A to the motive gravity be augmented in any ratio ; the proportion of AH to A I will be augmented in the same ratio, the latus rectum of the aboveAH2 mentioned parabola remaining the same, and also the length proporAI tional to it ; and therefore AH will be diminished in the same ratio, and AI will be diminished in the duplicate of that ratio. But the proportion of the resistance to the weight is augmented, when either the specific grav ity is made less, the magnitude remaining equal, or when the density of the medium is made greater, or when, by diminishing the magnitude, the resistance becomes diminished in a less ratio than the weight. RULE 4. Because the density of the medium is greater near the vertex of the hyperbola than it is in the place A, that a mean density may be preserved, the ratio of the least of the tangents GT to the tangent AH ought to be found, and the density in A augmented in a ratio a little greater than that of half the sum of those tangents to the least of the tangents GT. RULE 5. If the lengths AH, AI are given, and the figure AGK is to be described, produce HN to X, so that HX may be to AI as n -\- 1 to 1 ; and with the centre X, and the asymptotes MX, NX, describe an hyperbola through the point A, such that AI may be to any of the lines VG as XV" to xr. RULE 6. By how much the greater the number n is, so much the more accurate are these hyperbolas in the ascent of the body from A, and less accurate in its descent to K ; and the contrary. The conic hyperbola keeps a mean ratio between these, and is more simple than the rest. There fore if the hyperbola be of this kind, and you are to find the point K, where the projected body falls upon any right line AN passing through the point A, let AN produced meet the asymptotes MX, NX in M and N, and take NK equal to AM. RULE 7. And hence appears an expeditious method of determining this hyperbola from the phenomena. Let two similar and equal bodies be pro jected with the same velocity, in different angles HAK, h A k, and let them fall upon the plane of the horizon in K and k ; and note the proportion of AK to A A". Let it be as d to e. Then erecting a perpendicular A I of uny length, assume any how the length AH or Ah, and thence graphically,