252 On some Properties of Curves of the Second Order. 



Let j/^ =■ px + qx'^he the equation to any conic section, the 

 coordinate axes including any angle, but being such that the 

 axis 1/ coincides in direction with the line V M. q r. The equa- 

 tion to the tangents A B, A C drawn from the point A, (a, ^) is 



p^(jr— «)^— 4 {^x—uy) [p (j/ — jS) — 5'(/3x— aj/)] = 



and the equation to the chord B C is 



x{p-\-2qoi.)— 2^y-\- up = 



If these equations be transferred to the origin M, by put- 

 ting x + x' for X and 3/ + 7/ for y, x' and 3/' being the coordinates 

 of the point M,M.q and M r will be the roots of the first equa- 

 tion, considering y as the unknown quantity, when x is made 

 = 0, similarly M P is the value of j/ in the second equation 

 when X is made = ; therefore since by the theory of equa- 

 tions the product of the roots of y is equal to that part of the 

 equation which is independent of ?/, 



■Mr -MT p«(i'-a)--4(^x'-«y'){;7(y-/3)-9(/3x'-«y')} 



MgxMr = —. ^ -r i 



and since y^ = px' + qx'\ the truth of the equation 

 ^ pa _ 5 . ^''+J'' i M J X M r is easily recognized. 



This theorem may be extended to curve surfaces of the se- 

 cond order. 



If « |3y are the coordinates of the vertex of the conical sur- 

 face which circumscribes the curve surface of which the equa- 

 tion is 



7i^p^x^ 4- m^p^y^ + m^r^z^ = m^ 71^ p^ the equation to the 

 plane of contact is 



71^ p^ a X + m^ p^ ^ y ■\- «- m^yz = Tn^Ti^p* 

 if PMgr be any straight line parallel to the axis y cutting 

 the curve surface in M the circumscribing cone, of which the 

 equation is 



{Ti^p'-p^ /3^-w' f) {x-uy + {m^p^-ni' y'-p' «') {y-^f 

 + {m^7f—m^u^-n^ ^''){z-yf + 2p^ u^{x—ct){y-^) 

 + 27n^ ^y{z-y) {y-^) + 27i^ ay {z-y) {x-a) = 0, 

 in q and r and the plane of contact in P 



PM^=»^( '"^^"-f'!- "'^^')x MgxMr. 



^2 /m_P — ^ ,"sT"^ ~} ''' evidently independent of j:.y 2, that is, 



of the position of the point M. 



XXXVII. On 



