1881.] On certain Geometrical Theorems. 21?> 



is to be a point of inflexion, we shall have — 



dx* dx dx* dx 



+ eM—ntf %L + ei d t--mie d l + ^el{l-n + 0)^1 + m Q-m + c) 

 cfo 3 c2a? die 2 c&c dx 



or substituting for I — (pi 2 , and dividing by y^, we have — 



dx dx 



+ 0<&—neO + eb^ — mbe + <j)(l — n + a)el + <j)(l — m + c)hl=0. 

 dx dx 



Again substituting I— (pp for (1 + 0)^, and reducing, we obtain 



dx 



the equation 



(0 + l<ft)(ae + cb—ne — mb) = . 



Hence, if ae + cb — ne — mb vanish, the origin will be a point of in- 

 flexion, whatever values we give to and ; hence the theorem is 

 true. 



(3.) From any point six tangents can be drawn to a curve of the 

 third order ; two of these are at right angles to one another, determine 

 the locus of the point. 



Substitute for y in the general equation of the cubic m(x — + 

 arrange the terms of the resulting equation according to powers of 

 (x) and form the discriminant, equate the discriminant to zero, and 

 we shall have an equation of the form — 



m 6 — am 5 -\-bm i — cm 3 + dm? — em +/= . 

 Let mr L -\-m 2 '{-ni s + m 4: -\-m 5 -]-mQ=a, 



m l m 2 + (^h + m %) ( m 3 + m 4 + m 5 + m e) + m 3 m 4 + m 3 m S + m S m Q m i m 5 



+m 4 m 6 +m 5 m 6 =&, 



m 1 m 2 (m s + m 4 4- m 5 + th 6 ) 

 + (m 1 + w 2 ) (m 3 m 4 +m 3 m 5 + m 3 w 6 + m 4 ra 5 + m 4 m 6 + m 5 ??^ 6 ) 

 + wi 3 ra 4 m 5 + m 3 m 4 m 6 + m s m-m 6 + m 4 7?i 5 m 6 = c, 



m 1 m 2 (w 3 ra 4 + w 3 m 5 -f ra 3 m 6 + m 4 m 5 + m 4 m 6 + ?n 5 w 6 ) 

 + (tt^ + w 2 ) (m 3 ?n 4 m 5 + m 3 m 4 ra 6 + ?^ 3 m 5 ?7z 6 -f m 4 m 5 m 6 ) + m 3 m 4 m 5 W2 6 = 

 m 1 m 2 ( w 3 w 4 m 5 + m 3 m 4 m fi + m 3 m 5 m 6 + m^m-mo) 

 + (w^ + w 2 ) (m 3 , m 4 m 5 W2g) = e, 



m^myni 3 m 4 m-wig =/. 



