APPENDIX. 307 



have only to choose for the which occurs in this case, and which, as we per 

 ceive, is to be sought between 7 50 and 10 27 . 



The root is in fact 



2 = r 59 30&quot;.3, 

 and the remaining roots, 



&quot; = 26 24 3 



if* = 148 2 35 

 ^=18740 9 



are all found within the limits of the table. 



2. In the case of the fifth comet of 1847, Dr. GOULD derived from his first 

 hypothesis the equation 



[9.7021264] sin 4 * = sin (s -f- 32 53 28&quot;.5). 

 He had also 



V = 133 31&quot;. 



Then we have sin q &amp;lt;^ -f , and the inspection of the table shows that the factor 

 in the parenthesis lies between m and m&quot; ; therefore, there will be four real roots. 

 of which three will be positive. The given d approximates here most nearly to 

 s ia , about which, at any rate, there can be no doubt. 



Consequently, the paradoxical case of the determination of a double orbit 

 occurs here, and the two possible values of s will lie between 



88 29 - - 105 59 

 and 



105 59 -131 7 

 In fact, the four roots are, 



2 = 95 31 43&quot;.5 

 z&quot; == 117 31 13 .1 

 z ra = 137 38 16 .7 

 a&quot; =329 58 35 .5. 



By a small decrease of m without changing q, or by a small decrease of q 

 without changing m, a point of osculation will be obtained corresponding to 

 nearly a mean between the second and third roots ; and on the contrary, by a 

 small increase of m without changing q, or a small increase of q without changing 

 m, a point of osculation is obtained corresponding to nearly a mean between the 

 first and second roots. 



