70 



PROFESSOR CAYLEY ON POLYZOMAL CURVES. 



Investigation of the Foci of a Conic represented by an Equation in Areal Co-ordinates — 



Art, Nob. 166 to 109. 



166. I premise as follows : Let A, B, C be any given points, and in regard to 



the triangle ABC let the areal co-ordinates of a current point P be u, v,w. 



that is, writing PBC, &c, for the areas of these triangles, take the co-ordinates 



to be 



u : v : w a PBC : 1'CA : TAB , 



or, what is the same thing in the rectangular co-ordinates {x, y, z = 1), if 



(a,a',l),(b,b',l),(c,<S,l), 



be the co-ordinates of A, B, C respectively, take 



i> : v : vj = 



' , //, z 



: 







a , //, z 



b, V, 1 





'• '', 1 





". "', 1 



c, c, 1 





". »', 1 





b, V, 1 



or in the circular co-ordinates (£, >7, * = 1), if («,«', 1), (& /3', 1), (<y, </, 1) be the 

 co-ordinates of the three points respectively, then 



v : 



) : to — 



& P, 1 

 y, y, 1 





7- 7> 1 

 a. a . 1 





a. a, 1 



a /s-, i 



167. For the point / we have (£, >/, z) = (0, 1, 0), and hence if its areal co- 

 ordinates be (m , z> , w q ), we have 



tt : r : w, = ,3 — 7:7 — «:a-/3, 



and hence also, {u, v, m) referring to the current point P, we find 



%W - W 9 V = (7 - «)[(<*' - /3')(| - «z) -(«_£)(,,_ «'«)] 



- (a - £)[( 7 ' - a')(g - a:, - (y- «)(»»- *z)] = ft(g - »z) , 



if 

 whence 



n = 7 _«) la '-/3')-(a- /3)(y'-<0 





1 



/?,/?', 



1 



y, y', 



1 



v n w 



— w v : tc u — ira : u Q v — w = g — aZ : % — /& : £ — yz , 



and in precisely the same manner, if u \ v ', w ' refer to the point J, then 



tc : v : w' = P — 7 : y — a : a — /3' , 

 v' w — w' v : iv' u — «n*' : ii' i> — ia-' = *j — a!z : »j — S'z : % — y's . 



and 



168. Consider the conic 



(a, b, c,f, g, ]i)(u, v, wf = , 



