270 THE MATHEMATICAL PRINCIPLES [BOOK II. QQ __ 4RR Q.E.I. COR. 1. If the tangent HN be produced both ways, so as to meet any HT ordinatc AF in T - will be equal to V/T+ QQ; and therefore in what has gone before may be put for ^ \ -\- QQ. By this means the resistance will be to the gravity as 3S X HT to 4RR X AC ; the velocity will be a* •r-pj — --^, and the density of the medium will be as „— -TT-n. AO -v/ ±i Jti X H 1 COR. 2. And hence, if the curve line PFHQ be denned by the relation between the base or abscissa AC and the ordinate CH, as is usual, and the value of the ordinate be resolved into a converging series, the Problem will be expeditiously solved by the first terms of the series ; as in the fol lowing examples. EXAMPLE 1. Let the line PFHQ, be a semi-circle described upon the diameter PQ, to find the density of the medium that shall make a projec tile move in that line. Bisect the diameter PQ in A ; and call AQ, n ; AC, a ; CH, e ; and CD, o ; then DI2 or AQ,2 — AD 2 = nn — aa — 2ao — oo, or ec. — 2ao — oo ; and the root being extracted by our method, will give DI = e — ao oo aaoo ao3 a3 o3 ~e~~~2e 2e? ~~~ W ~2? ' &C* Here put nn f°r ee + aa> and ao nnoo anno3 DI will become = e — , &c. e 2e3 2e5 Such series I distinguish into successive terms after this manner : I call that the first term in which the infinitely small quantity o is not found ; the second, in which that quantity is of one dimension only ; the third, in which it arises to two dimensions ; the fourth, in which it is of three ; and so ad infinitum. And the first term, which here is e, will always denote the length of the ordinate CH, standing at the beginning of the indefinite quantity o. The second term, which here is — , will denote the difference between CH and DN ; that is, the lineola MN which is cut off by com pleting the parallelogram HC DM; and therefore always determines the CM? position of the tangent HN ; as, in this case, by taking MN to HM as — G to o, or a to e. The third term, which here is -£—, will represent the li neola IN, which lies between the tangent and the curve ; and therefore determines the angle of contact IHN, or the curvature which the curve line