318 James Elliot on certain 



the same times, if obeying gravity alone. Since these forces, 

 then, are in opposite directions, the resultant force will be 

 equal to their difference, and the resultant motion equal to 

 the difference of the motions which those two forces would 

 produce in the said times. There will, therefore, be a rise 

 or a fall of the lowest point according as the perpendiculars 

 above the horizontal line, or those below, are the greater. 



It is, however, the first pair of these perpendiculars — that 

 is, the nearest to the point of contact — which determines- 

 the resulting motion : if the first perpendicular, or abscissa, 

 of the parabolic curve be greater than the corresponding 

 abscissa of the ellipse, the lowest point will descend still 

 lower, and the top will fall ; but, if less, the lowest point will 

 attain a higher place, and the top will rise towards an upright 

 position.* Now, since the form of the ellipse, corresponding 

 to a given inclination of the top, is constant, while that of 

 the parabola widens or contracts as we increase or diminish 

 the velocity, it is evident that such a velocity may be given 

 to the top that any abscissa of the parabolic curve shall be- 

 come less than the corresponding abscissa of the ellipse, and 



* The tendency to rise or to fall (or rather the excess of tendency in favour 

 of a rise or of a fall) will never cease (the velocity of rotation being constant), 

 but will continue to urge the top either to rise towards a vertical position, or 

 to fall to the ground. For, A B being the same circumference which we have 

 supposed throughout, and C P the axis of the top, let the angle of inclination of 

 the top vary : the abscissae of the parabolic curve 

 above described vary as the force downward (or /; 



tendency to fall), and this varies as the sine of the I i 



angle of inclination, P C F. The abscissa? of the el- — -— ^/ ! 



lipse vary as the conjugate axis, — that is, as E B ; ' C^^^ /^v. "■ 

 and E B varies as the sine of the angle E A B, or >». T^-J ^\ ' 



P C F. Therefore the abscissae of the parabola vary ^^"^7^^ i "*^J B 



a9 those of the ellipse. Consequently, if the advan- / 



tage is in favour of either in any one position of the / 



top, it will continue so in every other; and if the top / 



begin either to rise or to fall, it will continue to do 



so, so long as the velocity remains unchanged. But though the ratio of the 

 said abscissae continues the same, their difference, when in favour of the ellipse 

 (which difference measures the preponderance of the upward force or of the 

 tendency to rise), will continually diminish. This difference will vary as the 

 sine of the angular distance remaining to be passed over, and ultimately as the 

 distance itself; therefore, I presume the axis will approach the vertical line 

 in an endless spiral, and will never attain an actually vertical position. 



