41 G Mr. A. Cayley on the Geometrical Representation 

 or under a somewhat more convenient form we haye 



> Vfl+A/ 



for the condition in order that the point (#, p', q 1 ) may lie in the 

 tangent plane at (k; p } q) to the surface (k). Similarly, we have 



z((b-c)(c-d)(d-b) S(a+p){a + q) ^(a+p^a + q 1 )-^!) =0 



for the condition in order that the point (k, p, q) may lie in the 

 tangent plane at (k!;p f } q') to the surface (k 1 ). The former of 

 these two equations is equivalent to the system of equations 



S(a+p)(a + q)(a+pf)(a + j)-!y^=* + f* + va*, 



V a-\-K 



and the latter to the system of equations 



V[a+p)(a + q){a+p^)(a + q t )\/^^=^-hA + y , a 2 ; 



• 



where in each system a is to be successively replaced by b, c, d, 

 and where X, p, v and X', p', v 1 are indeterminate. Now dividing 

 each equation of the one system by the corresponding equation 

 in the other system, we see that the equation 



x + k _ \ + px + vx 2 

 x + k'~~\ , + p'x + v , x' li 



is satisfied by the values a, b, c, d of x ; and, therefore, since the 

 equation in x is only of the third order, that the equation in 

 question must be identically true. We may therefore write 



X + px + vx 2 = (px + a) (x + k), X' + p'x + vV 2 = (px + a) (x -f k') , 



and the two systems of equations become therefore equivalent to 

 the single system, 



S{a+p)(a + q)(a+pF)(a + tf) = (pa + a) \/{a + k){a + k') 



V(b+p)(b + q)(b+pt)(b + q')=(pb + (r)\f(b + k)(b + k') 



* / (c+p){c + q){c+p')(c + q J ) = (pc + a)\f(c + k)(c + k!) 



\/(d+p)(d+q)(d+p f )(d+q f ) = (pd-r a) \/(d+k)(d+k!), 



a set of equations which may be represented by the single 

 equation 



yjr(x +p) (x + q) (x +p' ) (x^q J )-(px + a)\x + k){x + H) 

 =X(#-a)(ff-&)(*-c)(#-^ 

 where x is arbitrary; or what is the same thing, writing — # 



