•SEC. IY.1 OF NATURAL PHILOSOPHY. 291 COR. 9. And although these motions becoming eccentrical should be performed in spirals approaching to an oval figure, yet, conceiving the several revolutions of those spirals to be at the same distances from each other, and to approach to the centre by the same degrees as the spiral above described, we may also understand how the motions of bodies may be per formed in spirals of that kind. PROPOSITION XVI. THEOREM XIII. If the density of the medium in each of the places be reciprocally as the distance of the >, places from the immoveable centre, and the centripetal force be reciprocally as any power of the same distance, I say, that the body may revolve in a spiral intersecting all the radii drawn from that centre in a given, angle. This is demonstrated in the same manner as the foregoing Proposition. For if the centri petal force in P be reciprocally as any power SPn+1 of the distance SP whose index is n + 1 ; it will be collected, as above, that the time in which the body describes any arc PQ, i will be as PQ, X PS2U ; and the resistance in i!! x _ n;°raS "~' X SPn;PQ, X SP"XSQ,' , , 1 — in X OS . 1 — \n X OS . therefore as Qp"^~gpirqTT' tliat 1S> (because - ~~Qp~~ 1S a £lven quantity), reciprocally as SPn+ ! . And therefore, since the velocity is recip rocally as SP3", the density in P will be reciprocally as SP. COR. 1. The resistance is to the centripetal force as 1 — ^//. X OS to OP. COR. 2. If the centripetal force be reciprocally as SP3. 1 — ±w will be === 0 ; and therefore the resistance and density of the medium will be nothing, as in Prop. IX, Book I. COR. 3. If the centripetal force be reciprocally as any power of the ra dius SP, whose index is greater than the number 3, the affirmative resist ance will be changed into a negative. SCHOLIUM. This Proposition and the former, which relate to mediums of unequal density, are to be understood of the motion of bodies that are so small, that the greater density of the medium on one side of the body above that on the other is not to be considered. I suppose also the resistance, cateris paribus, to be proportional to its density. Whence, in mediums whose