APPENDIX. 305 



Then the following are the conditions upon which it is possible to find a 

 planet s orbit different from that of the earth, which shall satisfy three complete 

 observations. 



First. The equation 



m sin 4 z = sin (z -\- q) 



must have four real roots. The conditions necessary for this are, that we must 

 have, without regard to sign, 



sin q &amp;lt; | 



and m must lie between the limits m and m&quot;. 



/Second. Of these four real roots three must be positive and one negative. 



For this it is necessary that cos q should remain positive for all four of those 

 values for which 



sin q &amp;lt; , 



the two in the second and third quadrants are excluded, and only values between 

 - 36 52 and + 36 52 are to be retained. 



If both these conditions are satisfied, of the three real positive roots, one must 

 always correspond to the Earth s orbit, and consequently will not satisfy the 

 problem. And generally there will be no doubt which of the other two will 

 give a solution of the problem. And since by the meaning of the symbols, arti 

 cles 139, 140, we have 



sin z sin (8 z) sin & 



IT ~tf~ : ~7~ 



not only must z and d be always less than 180, but, also, sin(d z) must be 

 positive, or we must have 



y&amp;gt;a. 



If, therefore, we arrange the three real positive roots in the order of their 

 absolute magnitudes, there may be three distinct cases. Either the smallest root 

 approaches most nearly the value of d , and corresponds, therefore, to the Earth s 

 orbit, in which case the problem is impossible; because the condition d &amp;gt;2 can 

 never be fulfilled. Or the middle root coincides with d , then will the problem 

 be solved only by the smallest root. Or, finally, the greatest of the three roots 

 differs least from d . in which case the choice must lie between the two smaller 



39 



