292 Rev. T. P. Kirkman on Bisignal Univalent Lnaginaries. 



form of the involute of the circle whose radius is CN, and 

 centre C. The length of the curved portion of the ruler should 

 be somewhat more than one-half of that of the straight part. 



The use of this instrument is as follows: — 



Problem. — The elliptic orbit of a body, gravitating towards 

 a given focus, being given, to represent on a diagram the place 

 of the body corresponding to a given mean anomaly. 



Solution. — About a centre C with the radius CN describe 

 the circle PNQADP, and let its diameter AP be taken to 

 represent the major axis of the given elliptic orbit. Let S 

 represent the focus of attraction, P the perihelion, and the 

 angle PCN the given mean anomaly. Apply the anomaly- 

 ruler to the diagram as shown in the drawing, so that the mark 

 C shall be over the centre of the circle, the mark N at the 

 end of the arc of mean anomaly, and the convexity of the in- 

 volute NI towards the perihelion. From the focus S draw a 

 straight line ST touching the involute NL Through the 

 centre C draw a straight line CQ parallel to ST and on the 

 same side of AP. From the point Q, where CQ cuts the 

 circle, draw a straight line QB perpendicular to AP; divide 

 this line by the point M in such a proportion that BQ : BM : : 

 major axis : minor axis of the elliptic orbit. Then the point 

 M represents the place of the body corresponding to the mean 

 anomaly PCN. Q. E. I. 



The proof of this solution follows immediately from the 

 31st proposition of the first book of Newton's Principia. 



If the entire ellipse PMAEP, similar to the given orbit, can 

 be described with sufficient accuracy, the point M may be 

 determined by the intersection of the ordinate BQ with the 

 ellipse, without performing the proportional division of thatline. 



It has been suggested by the Astronomer Royal for Scot- 

 land, that by means of a turning-lathe a series of ellipses might 

 be engraved on a plate with the same major axis, varying in 

 excentricity between a circle and a straight line; and that 

 impressions of this plate might be used in the graphic repre- 

 sentation of cometic motions, with the aid of an anomaly-ruler 

 of suitable size. 



Glasgow, August 1850. 



XXXVI. On Bisignal Univalent Imaginaries, 

 By the Rev. Thomas P. Kirkman, M.A.* 



IF abcdefghiklmnop be fifteen imaginaries having no linear 

 relation to each other, and such that 

 a 2 =6 2 =..=p 2 =-.l, 

 and also that by definition every pair of them, as m and n 9 

 * Communicated by the Author. 



