TRANSACTIONS OF SECTION A. 465 



be described about the origin under the action either of the force 



fir 



or the force 



the periodic time will be 



2tt 



3 



{ax + by + c) 



fJT 



(A.z 2 + 2H.ry + By 2 )*' 



c(AB-H 2 ) 1? 



J P 1 AB - H 2 - (Aft 2 + Ba 2 ^2IL^) J '* 



3. In the case of an elliptic orbit described about any point 0, the law of force 

 to may be put in the form 



i*<F-PtP>)*' 



where o is the semi-axis minor ; r is the distance OP of the body P from : » , p 

 are the perpendiculars from the foci 8, H upon POP' (the chord through P and 0) 

 and p } ,p 9 have the s same or opposite signs according as S, H are on the same or 

 opposite sides of PP'. The periodic time is then 



2»r, , 



a being the semi-axis major. 



4. If a parabola be described about any point 0, the force tending to O, at any 

 point P in the orbit, is equal to p ^ where OM is drawn parallel to the axis 

 meeting the tangent at P in M. 



This, and the expression for the force in § 3, may be deduced from the 

 scholium to Prop. xvn. of Newton's ' Principia.' 



The Paper of which the above is an abstract will appear in the 'Monthly 

 Notices' of the Royal Astronomical Society. J 



11. Note on the Geometrical Treatment of Bitircidar Quartics. 

 By Frederick Purser, M.A. 



_ The author pointed out that some of the leading properties of these curves 

 might be readily deduced geometrically from the definition of the curve as a locus. 

 This definition may be thus stated: "The bicircular quartic is the locus of points 

 P , whose quasi polar, to a circle J touch a conic F" ; by quasi polar being under- 

 stood the parallel to the polar half way between point and polar 



-n^rt ^ th 7 6 CU '? Ula1 ' P ° ints w > w '> the Hcircular becomes the binodal quaitic, 



and the quasi polar the companion chord of intersection of the lines P«> Pa/ 

 with the conic J. ,.».<« 



ftl +WnS^f' th il meth ° d , y ! elds a geometrical proof of the theorem of 

 orthogonal section of two confocal quartics, and the same mode of proof is appli- 

 cable to the corresponding theorem for cyclides 



thro^Tti,fL POriS ? 0D * h « iM °"Ption of polygons whose sides pass alternately 

 through the two nodes of a binodal quartic is shown to be reducible to the porism 

 o the m-and-circumscnbed polygon for conies. Lastly, it is shown that this 

 mode of treatment leads to the following theorem, which is, so far as the author is 



!Sfc*. fv IT • w 0n A C U 1 t0Uch a Une L > a conic v » and ha ve double contact 

 with a third come W, the chord of contact envelopes a binodal quartic, of which 

 the intersections of L and W are the nodes." 



12. On Quadric Transformation. By Professor H. J. S. Smith. 

 1878. H H 



