274 term THE MATHEMATICAL PRINCIPLES [BOOK H nbb nn -f- n bb U ! J • x • £ • *° •=rr be resolved into an infinite series -r- + — A — Of A" A.n 3 -±- 3nn + 2/i X O + n " ~ x bb O3,&c,,andGD will be equal g^TT-T X 00 O2 + c£ bb d nbb + ?m toC — -A--T-+-O- -r O - ~ e A" e A" + l 2An -f + H i^T 't"\ bb°3> &c- The second tcrm - O - -T 6An + e An 4- l series is to be used for 0,0, the third ^ 0 66O2 for Roo, the fourth — -£\~r~3 bbO5 for So3. And thence the density of the medium Oof this -, in any place G7 will be 2dnbb nub*' and therefore if in VZ you take VY equal to n X VG, that density is ren w IT- j ^ * 2rf//66 /mfi4 ciprocally as XY. For A2 and — A2 -- — A + — r are the tc/ o^x ./\_ " squares of XZ and ZY. Hut the resistance in the same place G is to the force of gravity as 3S X - to 4RR, that is, as XY to And the velocity there is the same wherewith the projected body would move in a parabola, whose vertex is G, diameter GD, and latus rectum 2XY2 or - -------- - - . Q.E.I. R nn VG AC HT SCHOLIUM. In the same manner that the den sity of the medium comes out to be as S X AC . « Tjr™ m ^°r- 1) if the resistance lx X HI is put as any power V" of the velocity V, the density of the medium will come out to be as B C D E Q — . x S And therefore if a curve can be found, such that the ratio of to 4 — o R— i