305 APPENDIX. 



roots. Each of these will give a planetary orbit, because each one fulfils all 

 the conditions, and it will remain to be determined, from observations other than 

 the three given ones, which is the true solution. 



As the value of m must lie between the two limits m and m&quot;, so also must 

 all four of the roots lie between those roots as limits which correspond to m and 

 m&quot;. In Table IV a . are found, therefore, for the argument q from degree to degree, 

 the roots corresponding to the limits, arranged according to their magnitude, and 

 distinguished by the symbols z\ z&quot;, z m , z&quot;. For every value of m which gives a 

 possible solution, these roots will lie within the quantities given both for m and 

 ni&quot;, and we shall be enabled in this manner, if 8 is found, to discern at the first 

 glance, whether or not, for a given m and q, the paradoxical case of a double orbit 

 can occur. It must, to be sure, be considered that, strictly speaking, 8 would 

 only agree exactly with one of the z s, when the corrections of P and Q belong 

 ing to the earth s orbit had been employed, and, therefore, a certain difference 

 even beyond the extremest limit might be allowed, if the intervals of time should 

 be very great. 



The root s&quot;, for which sin s is negative, always falls out, and is only intro 

 duced here for the sake of completeness. Both parts of this table might have 

 been blended in one with the proviso of putting in the place of z its supplement ; 

 for the sake of more rapid inspection, however, the two forms sin (z &amp;lt;/) and 

 sin (.? -|- &amp;lt;/) have been separated, so that q is always regarded as positive in the 

 table. 



To explain the use of Table IV. two cases are added ; one, the example of Ceres 

 in this Appendix, and the other, the exceptional case that occurred to Dr. GOULD, 

 in his computation of the orbit of the fifth comet of the year 1847, an account of 

 wJiich is given in his AstronomicalJournal, Vol. I., No. 19. 



I. In our example of Ceres, the final equation in the first hypothesis is 



[0.9112987] sin 4 z = sin (*. 7 49 2&quot;.0) 

 and 



8 = 2419 53&quot;.34 



the factor in brackets being the logarithm. By the table, the numerical factor 

 lies between m and m&quot;, and this d answers to z&quot;, concerning which there can be 

 no hesitation, since z u must lie between 10. 27 and 87 34 . Accordingly, we 



