ON TBILINEAR CO-ORDINATES. 63 



Let B'C' = p', C'A' = g', A'B'=r', and letp, q, r be the perperfdie- 

 ulars from A, B, C on the line A'B'C ; then 



sin B sin A sin A 



p' sin (4 sin -i^ 



p = AB' sin rt = —-!—-» 



sin ^ 



, ,„ . (7^ sin li/sln 9 



o = ^^5 sin e ^ '^ :--7T , 



sin B 



. ,r, . r. r' sin sin* 

 7- = ^^Csin0^ ^ 



sin (J 



pf » sin yl 



* • _i. -^ otC. 



" sin sin sin sin ^' 

 whence from (2) it follows that 



I m n 



(3) 



/> sin J. g sin 5 r sin C 



3. From (2), since B' 0' ^ C A! -\- A!B' ^^, there, follows the 

 relation 



Z sin -f- ™ sin ^ 4- w sin >// = (4) 



the signs being determined by the convention that lines measured in a 

 direction contrary to the first named (as B' C) are considered negative. 

 The relation (4) can also be deduced immediately from the two 

 forms of the equation to the line 



a — f — 9 y — h 



Za_Lm/34-wv = 0and -^^='-^-^ = ^-r-r 



' 1^ > ' sin sin sin \p 



4. To find the angles made by the line la -\- mj3 -{- ny = with the 

 sides of the triangle of reference. 



In (4) 6, <p, 4> are always <; -k ; but that relation may also be written 

 in the equivalent form 



Z sin + w sin -I- " sin := 0, 



the previous convention being neglected, if <?i, 6.^^ 6.. be the angles 



which the sides of the triangle of reference when projected make with 



the given line. 



Projecting a, 6, c on a line perpendicular to the given line we get 



a sin + 6 sin +csin0 =0 

 1 ' 2 ' 3 



Bin 0j sin 0^ Bin 0, 



bn — era cl — an am — bl 



j 2 sin ^ sin 5 sin C 1 ^ 

 ~ I ibn — cmY sin 2 4 + ) .... (5) 



