378 PHILOSOPHICAL TRANSACTIONS. [aNNO I767. 



— —, the lesser axis of the curve would then vanish, and the cusps meet in the 



centre, and form two distinct shadows, as represented in fig. l6; in consequence 

 of which, if a satelles, revolved at that distance, it might suffer a double eclipse, 

 at the same conjunction, which remarkable phenomenon may also happen, at a 

 less distance from Jupiter, in some circumstances. Mr. W. now shows how the 

 duration of an eclipse of a given satelles may be determined independant of the 

 equation of the curve ; and this perhaps will be the more acceptable, as it affords 

 a practical rule, which may be applied, in every position of Jupiter's axis, with 

 very little trouble. This may be done by the help of the following proposition. 



Prop. 6. — If a circle enjo, fig. 17» be described about the conjugate axis gd, 

 of a given ellipsis adbg, and a right line ef be drawn, making the given angle 

 Fttd, with that axis, and passing through the given point tt taken in it; it is 

 proposed to determine the length of the segments vf, Ee, intercepted between 

 the circumference of the circle, and the perimeter of the ellipsis. 



From the point f, draw the right line rd parallel to the transverse axis ab, 

 meeting the conjugate gd in the point d, and the circle in c; draw the lines of, 

 cf, cc, and let c-n- be joined: then, by conies, as cb : cd :: tang, fwd : tang. cirD, 

 and in the right lined triangle Cnc, as cc (cd) : sin. Cttc :: Cv : sin. cctt, whence 

 the angle cCtt becomes known; but as cd : cb :: tang, cctt : tang, fctt; therefore 

 fctt is known; from which taking away the given angle /ctt, there remains the 

 angle fcf; consequently all the angles, in the right lined triangle /cf, together 

 with the side c/(cd), are known: we shall therefore have, in the right lined 

 triangle, f/c, as sin./FC : cf-.ism.fcF i/f, one of the required segments; and 

 by a similar operation, the other segment Ee will be found, whence as ef'is given, 

 EF will become known. 



Carol. 1. The required segments f/", Ee, will be found in the same manner, 

 when the given point n is not taken in one of the axes, but any where between; 

 but in that case, the point where the line ef intersects the conjugate axis, must 

 be first determined. 



Carol. '1. If a perpendicular en be let fall from c on the line ef, the angle 

 ttCw will be given, to which adding fctt (found above) the angle Fcra will be 

 known; hence we shall have the following analogy for determining fw; as tang. 

 fen: tang. ¥cn::/n : p«. Now let k^h, fig. 18, represent the disk of the sun, 

 and ep/G that of Jupiter, considered as a circle, whose diameter is equal to his 

 axis dg: draw n/jji, the path of the satelles, making the given angle n/jr, with 

 a right line Rg drawn parallel to the diameter dg, and let ab be the duration of 

 the eclipse, and v the apex of the shadow in this hypothesis; join va, v^, and 

 let the plane avb be produced, till it meets the sun's disk in k and k, it will then 

 intersect the disk of Jupiter in the lineyire, and the lines vk, vh will also touch 



