PHILOSOPHICAL SOCIETY OP WASHINGTON. 61 



substituting in (10) we find after reducing 



w* + n^ + (V 8 '^ — 1 ) n, — 1 = 1 

 ^ a -^ 1 



/ \ r (10 



m' + wi'^ + (n/ 8 * — 1 )m — I = 0. 



It is to be noted that n ^ = tan ^ B and m = = tan 



a b 



^ A, and therefore Eq. (11) corresponds to Eq. (2). 



FIFTH SOLUTION. 



The fundamental relations between the sides and bisectors are 



bc(a + b + c)(-a+b + c) bo 



{b + cy ^^^'^^'^^''—''\b+c)t 



., ac (a + b -4- c) (a — b + c) , ac 



»' ="^ (^r -^' = (a- + 2 ao + .' _ 6.) ^-^^^ 



And since a^ + b^ = g* 



a' = 2 ¥ or • = — ■ — = 14--, 



b -{-C a^ C 



^^ = 2a» -^- or ^'=^+-^:= 1 + -^ 

 "Whence 



-as in the second solution, where this relation was obtained trigo- 

 nometrically. Again 



cM 



2 JQ' + g^ + & ^ a + & l ^ o' + 2 a6 4- 6^ ^ a+ 6 j 

 Again by adding 



2/2a6_o+6 

 oi3 c 



Whence 



2«: , 2j-^__^i+i , 2 



/3» ^ a^ — C ^ ■ 



2a^ 2N/2a6 , 26» , 



-^ — ^ f- — = 1 (4) 



p op a 



as previously obtained trigonometrically. The solution i.s now 

 •completed as in the second solution. 



