304 APPENDIX. 



Contact of the curves can exist only when for a given value of q, 



z 1 = i q -j- i sin&quot;&quot; 1 f sin q 

 and 



, sin (/ q) 



sin 4 z / 



If the contact of the curve of the fourth order with the sine-curve is with 

 out the latter, then will m constitute the upper limit, for m greater than this 

 values of the roots will be impossible. There would then remain only one positive 

 and one negative root. 



If the contact is within the sine-curve, then will the corresponding m&quot; con 

 stitute the lower limit, and for m less than this, the roots again would be re 

 duced to two, one positive and one negative. 



If q is taken negative, or if we adopt the form 



m sin 4 2 = sin (z -j- q) 



180 - z must be substituted for z. 

 The equation 



m 2 sin 8 z 2m cos q sin 6 z -j- sin 2 z sin 2 q =. 



shows, moreover, according to the rule of DESCARTES, that, of the four real 

 roots three can be positive only when q, without regard to sign is less than 

 90, because m is always regarded as positive. For q greater than 90, there is 

 always only one real positive root Now since one real root must always cor 

 respond to the orbit of the Earth, that is, to r = R ; and since sin&amp;lt;5&quot;, in the 

 equation, article 141, 



R sin & 

 sin z = -; 



is always positive, so that it can be satisfied by none but positive values 

 of z ; an orbit can correspond to the observations only when three real roots are 

 positive, or when q without regard to its sign is less than 90. These limits are 

 still more narrowly confined, because, also, there can be four real roots only 

 when m lies between m and m&quot;, and when we have 



| Bin q &amp;lt; 1, or sin q &amp;lt; f , q &amp;lt; 36 52 11&quot;.64 

 in order that a real value of / may be possible. 



