14 



MEMOIK.S OF THE NATIONAL ACADEMY OF SCIENCES. 



the axes of d and h respectively. The vanishing points will be on the axis OH at distances h' and 

 It" above and below O. Upon this diagram are shown the halves of four iseijgoual ellipses. The 

 scales used for d and h are not equal to each other, since the use of the same scale for both coor- 

 dinates w(;uid make the figures of inconvenient shape. In this, and in all the Figs. 2-18, the unit 

 in d is to the unit in 7;, as 1 to sin oo. But to indicate more deaily this scale, and at the same 

 time to give a kind of shading to a part of the area, there are drawn above the radical axis ae lines 

 parallel to OE and parallel to OH, at intervals of -01; that is, the sides of each of the small rec- 

 tangles in the quadrant HOE are -01, or about 925,000 miles. Only the positive values of d are 

 represented in the figures. The positive vanishing point being 1*250 of these divisions above O, 

 and the negative vanishing point 15.174 below O, we layoff 0«=^(/t' -|-fe")=— 6-962 divisions, 

 and draw ae for the radical axis. The smallest positive value of ® is (Table ii) 3-04. As ® in- 

 creases from 3-04 the ellipse increases in size, and the innermost curve represents Mhat it becomes 

 when ®=5. The second curve (separating the blank and shaded areas) corresponds to ® = 20. 

 Any parabolic comet passing Jupiter with an original angle of o. =10°, and having d and h such 

 as to be represented by a point within the blank area of Fig. 2 will leave the vicinity of the planet 

 in an elliptic orbit whose semiaxis major is less than 20, and whose period, therefore, is less than 

 ninety years. 



Fig 4 5 



The larger curve that lies above' ae in the shaded area is the isergonal ellipse for 'g =50. 

 As -S increases the lower part of the curve tends to approach the radical axis ae, with which it 

 coincides when 'a— <x. For points iji the area below ae (distinguished by the oblique-line shading), 

 the planet increases the velocity of the comets, and the comet would l)o tlnown i)ermanently out 

 of the solar system. The smallest semitransverse axis, the one coii csponding to the vanishing 

 ellipse is (Table ii) 36-90, and the isergonal curve for ^^-50 is drawn in the figure. 



23. Isergonal ellipses for oj—ll{i°. — In Fig. 3 are drawn the three ellipses corresponding totlie 

 values of ®, —5, —20, and —50. The ellipses above ae do not appear, inasmuch as the smallest 

 possible elliptic orbit has a semiaxis major of £06-3 (Table ii), and a period of about 3,000 years. 

 The radical axis ae is -08146 (or over 8 divisions) above OE. 



24. Figures 4 and 5 are like diagrams for oj = 20° and od — 160°. With altered numbers tlie 

 cxi)lanations of arts. 22 and 23 apply with slight change to these figures. The line ae in Figs. 4 

 and 5 is nearer to OE than is the same line in Figs. 2 and 3. In Fig. 4 the line for -2.= —20 appears 

 below «p, while above ae are the three curves for + "', +20, and +50,respectively. In Fig. 5 the ellipse 

 for ®=.50 is wanting since the minimum ellipse has a semiaxis major 52-36 (Table ii), while below 

 '•(■ the three curves are present. 



