688 PHILOSOPHICAL TRANSACTION'S. [aNNO \7q6. 



the axis inclined to the vertical line. These limits agree precisely with those 

 which are demonstrated hy Archimedes, in the 2d book, of his tract, intitled 

 De iis quEC in humido vehuntur *, prop. 2 and prop. 4. If the specific gravity 

 of the parabolic conoid should be less than the limit which has just been investi- 

 gated, and if the axis should be to the parameter in a proportion greater than 

 that of 3 to 4, and less than that of 1 5 to 8, it will float permanently on the 

 fluid with the axis inclined to the horizon, and with the base wholly extant above 

 the surface at some angle less than 90°. 



And again : various inferences follow from this determination. In the first 

 place, though the object of the preceding investigation was, to find a single value 

 only of the specific gravity, which would cause the solid to float permanently 

 with the extremity of the base coincident with the fluid's surface, yet by the re- 

 sult it appears, that there are 2 values of the specific gravity which will 

 answer this condition under a certain limitation, which is also discovered by the 

 solution ; this is, that the axis (a) shall be to the parameter (p) in a proportion 

 greater than that of J 5 to 8 ; for if that proportion should be less, 8a will be 

 less than ]5p; in which case the quantity y'Sa — 15/) becomes impossible. 

 From which circumstance it may be inferred, that whenever the axis is to the 

 parameter in a less proportion than of 1 5 to 8, the solid will float permanently 

 on the fluid with the whole of the base extant above the fluid's surface, whatever 

 may be the specific gravity of the solid. This limit is precisely the same with 

 that which is demonstrated by Archimedes, in the 2d book of his tract, intitled 

 De iis quae in humido vehuntur, prop. 6. When the axis bears a greater pro- 

 portion to the parameter than that of 15:8, the solid will float either with the 

 base entirely out of the fluid, or partly immersed under it, according to the 

 specific gravity. Having given the axis a in a greater proportion to the para- 

 meter /> than 15 to 8, by making the specific gravity 



_f26a-l5p + 6 X ^■2ax V{Sa — l5p) ^ci _ . 20"a - 15p — 6 x -v/2a X ^/ (8a— 15^) >a 



"— V 300 ) or7i — \^ ^y^ ;, 



* The demonstrations of" jVrchimedes, which relate to tlie paraboHc conoid, are founded on a sup- 

 position that this solid is generated by the )-evolution of a rectangular parabola on its axis ; that is, of 

 a parabola which is the section of a rectangular cone; in which case the line, called by the author 

 (or rather by his translator, the original of tliis treatise being lost) " ea quae usque ad axem," is half 

 the principal parameter, being equal to the perpendicular distance between the plane which touches 

 the cone, and the plane parallel (o it, which is coincident witli the parabola. T his solid is termed by 

 Archimedesj " conois rectangula :" but the limitation appears to be unnecessary, because the demon- 

 strations of the author are equally applicable to a solid generated by tjje revolution of a parabola 

 which is the section of any cone, whatever may be the angle at tlie vertex, half the parameter being 

 substituted instead of the line called by Archimedes " ea quae usque ad axem;" and it is a property 

 fof conies easily demonstrable, that any parabola being given, a similar and equal parabola may be 

 ormed from the section of any cone, whatever may be tlie angle at tlie vertex, the axis being of suf- . 

 ficient length. 



