VOL. v.] PHILOSOPHICAL TRANSACTIONS. 427 



make the same right line with bp. And since the same demonstration holds of 

 any other intersection of lines drawn from h and h, to the corresponding points 

 of the true and mean anomalies; it is plain, that not only the right line that 

 joins these intersections, will pass through the point b, but that hb will be a 

 perpendicular to it, Q. E. D. 



Corol. If from any point of the true anomaly, as A, to a corresponding point 

 i of the mean anomaly, we draw the right line h i; then ^y raised perpendicular 

 to cbd will cut h i, in s, in the ratio which the line of the mean motion has to 

 that of the true. For, by the latter analogy of the first corollar}', kbis the half 

 subtense: consequently, by cor. 2, the perpendicular erected from bj viz, b t, cuts 

 the diameter h h, in t, in the same ratio that the line of the mean motion has to 

 that of the true. Therefore r s, or bf, cuts h i in the same ratio in j, because 

 of the similar figures tb hkphb and s rhi ghr. 



Another method of finding the apogee and eccentricities from that of Dr. 

 Ward, for finding the first inequality, is thus: Let / and c^ be the two foci of 

 the ellipsis, (fig. 5) ^ and u two points of the planets true motion ; ^w an arch 

 of the ellipsis, seen from / under the angle tlu, and from d under the angle tdu\ 

 also Id, the distance of the foci, seen from t, under the angle dtl, and from w, 

 under the angle dul. I say, the difference of the angles tlu and tdu, is equal to 

 the difference of the angles dtl and dul. 



For since the sum of the three angles of the triangle lux is equal to that of 

 the triangle dtx; and if from both sums, the equal angles Ixu and dxt be de- 

 ducted, the remainder will be ulx -J- lux = tdx + dtx\ and if from these two 

 sums be taken the unequal angles w/>r and tdx, the difference of the remaining 

 angles lux and dtx, is equal to the difference of the subtracted angles m/jc 

 and tdx. 



With the centre /, and distance mn of the transverse axis, describe the circle 

 abc, whose arch ab i^ seen from / under the angle alb, and from d under the 

 angle adb. Also the distance of the foci Id is seen from a under the angle lad, 

 and from b under the angle Ibd-, therefore again the difference of the angles 

 alb and adh, is equal the difference of the angles lad and Ibd. But, by cor. 1, 

 the angle lad'is half the angle /mc?, and the angle Ibd, half the angle, ltd; 

 therefore the difference of the angles /ac/and Ibd, is equal to half the difference 

 of the angles lud and ltd; and consequently, the difference of the angles alb 

 and adb, is equal to half the difference of the angles ult and udt, the fornier 

 of which is the apparent interval of the two observ^ations, and the latter, the in- 

 terval of the mean motion. The difference of these intervals being given, we 

 have the half of this difference, viz. the difference of the angles alb and adb, 



3h2 



