Page:Newton's Principia (1846).djvu/297

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 performed 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 centripetal 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, will be as PQ, $\scriptstyle \times$ PS^½n; and the resistance in P as $\scriptstyle {\frac {Rr}{PQ^{2}\times SP^{n}}}$, or as $\scriptstyle {\frac {{\overline {1-{\frac {1}{2}}n}}\times VQ}{PQ\times SP^{n}\times SQ}}$, and therefore as $\scriptstyle {\frac {{\overline {1-{\frac {1}{2}}n}}\times OS}{OP\times SP^{n+1}}}$, that is (because $\scriptstyle {\frac {{\overline {1-{\frac {1}{2}}n}}\times OS}{OP}}$ is a given quantity), reciprocally as SPⁿ⁺¹. And therefore, since the velocity is reciprocally as SP^½n, the density in P will be reciprocally as SP.

Cor. 1. The resistance is to the centripetal force as $\scriptstyle {\overline {1-{\frac {1}{2}}n}}\times OS$ to OP.

Cor. 2. If the centripetal force be reciprocally as SP³, 1 - ½n 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 radius SP, whose index is greater than the number 3, the affirmative resistance 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, cæteris paribus, to be proportional to its density. Whence, in mediums whose