;;08 APPENDIX. 



&quot;We have, therefore, the choice between the two orbits. The root used by Dr. 

 GOULD was z&quot;, which gave him an ellipse of very short period. The other obser 

 vations showed him that this was not the real orbit. M. D ARREST was involved in 

 a similar difficulty with the same comet, and arrived also at an ellipse. An ellipse 

 of eighty-one years resulted from the use of the other root. 



&quot; Finally, both forms of the table show that the exceptional case can never 

 occur when 8 &amp;lt; 63 26 . 



&quot; It will also seldom occur when d &amp;lt; 90. For then it can only take place 

 with the first form sin (s q), and since here for all values of q either the limits 

 are very narrow, or one of the limits approximates very nearly to 90, so it will 

 be perceived that the case where there are two possible roots for d &amp;lt; 90 will 

 very seldom happen. For the smaller planets, therefore, which for the most part 

 are discovered near opposition, there is rarely occasion to look at the table. For 

 the comets we shall have more frequently d &amp;gt; 90 ; still, even here, on account 

 of the proximity to the sun, d &amp;gt; 150 can, for the most part, be excluded. Con 

 sequently, it will be necessary, in order that the exceptional case should occur, 

 that we should have in general, the combination of the conditions d ^&amp;gt; 90 and 

 &amp;lt;l between and 32 in the form sin (z q), or between 22 and 36 52 in the 

 form sin (z -j- q].&quot; 



Professor PEIRCE has communicated to the American Academy several methods 

 of exhibiting the geometrical construction of this celebrated equation, and of 

 others which, like this, involve two parameters, some of which are novel and 

 curious. In order to explain them, let us resume the fundamental equation, 



m sin 4 g = sin (z q). 



1. The first method of representation is by logarithmic curves ; the logarithm 

 of the given equation is 



log m -\- 4 log sin z = log sin (2 ). 

 If AVC construct the curve 



y = 4 log sin z, 



