368 Mr. J. J. Sylvester on a simple Geometrical Problem. 



and for the same reason as before, if v lies between co and 1, 

 tbis equation cannot be satisfied. Hence the theorem is proved 

 to be true for all values of n, except between 4- 1 and — 1 . For 

 these values it ceases to be true ; in fact, for such values for any 

 given values of {et—^) there will be always, as it may be easily 



proved, one or more values of (« + ^) ; thus if n = ^, the equa- 

 tion becomes 



tan 3 



, o — > 



andifn=-^, 3«-/3 



tan B- = l. 



« — p 

 tan — ;^ — 



showing that« + yS = 90and«— ^= +90 in these respective cases 

 will afford a solution over and above the solution « = /3, which is 

 easily verified geometrically*. It would be an interesting inquiry 

 (for those who have leisure for such investigations) to determine 

 for any given value of ?j between +1 and —1 the superior and 

 inferior limit to the number of admissible values oi a, + /3 cor- 

 responding to any given value of a— ySf- 



My reader wll now be prepared to see why it is that all the geo- 

 metrical demonstrations given of this theorem, even in the sim- 

 plest case of all, viz. when ?i=2, are indii-ect, I believe I may 

 venture to say necessarily indirect. It is because the truth of 

 the theorem depends on the necessary non-existence of real roots 

 (between prescribed limits) of the analytical equation expressing 

 the conditions of the question ; and I believe that it may be 

 safely taken as an axiom in geometrical method, that when- 

 ever this is the case no ether form of proof than that of the 

 reductio ad absurdum is possible in the nature of things. If this 

 principle is erroneous, it must admit of an easy refutation in 

 particular instances. 



As an example, I throw out (not a challenge, but) an invita- 

 tion to discover a direct proof, if such exist, of the following 



* lu the first of these cases, if the base of the triangle is supposed given, 

 the locus of the vertex is a right hue and a cu'cle ; in the second case, a 

 right hne and an equilateral hj-perbola. 



t When +n lies betw een . _ and - — - (( being any positive integer), 



it is easily seen that the superior hmit must be at least as great as (t). 



