474 The Rev. Geoege Salmon on the Degree of the 



drawn through the line joining the two fixed points. We shall suppose the 

 point xyzw to be on the surface, and the point .TiT/iZiWj to be taken anywhere 

 on the tangent plane at that point ; then we shall have U= 0, AiU — 0, and 

 the discriminant will become divisible by the square of A^U. For plainly, of 

 the tangent planes, which can be drawn to a surface through any tangent line 

 to that surface, two will coincide with the tangent plane at the point of contact 

 of that line. If the tangent plane at wyziv be a double tangent plane, then the 

 discriminant will be divisible by the cube of Aj U. If we write, for brevity, 

 Ai^U= A, A,A2U=B, A.^U= C, so as to make the coefficient of \"-^ in [^7] 

 to be written Afr + 2Bij.p + Cv^, then I say that the coefficient of the square of 

 Aa^Jin the discriminant of [f7] is 



A(B'-ACyc=i, 

 where a is the discriminant of the equation when U = 0, AiU^O, A2f'^= 0. 

 I have verified this in the case of the equation of the third degree, and I feel 

 that I am safe in asserting it to be true in general. In order, then, that the 

 discriminant should be divisible by the cube of A2U, some one of these fiictors 

 must either vanish or be divisible by A2U. 



First, then, let A =: 0, or Ai^U = 0. This will be the case if the point 

 x'y'z'w' be taken on either of the lines which can be drawn through xyzw so 

 as to meet the surface in three consecutive points. We shall suppose, however, 

 that the point x'y'z'w' has not been so assumed, and then, as A does not con- 

 tain x^y^z-iuo^^ this factor may be set aside as irrelevant to the present dis- 

 cussion. 



Secondly, let B- — AC be divisible by Aj U. Let it be required to find 

 the condition to be satisfied by the point xijzw in order that this should be the 

 case. Now '\{ B^ — AC contain A2 ?7 as a factor, any arbitrary right line which 

 meets the plane* Ajf/, will meetiJ^ — ^C; and, therefore, if we ehminate 

 x^y^ZiW^ between these two equations and those of an arbitrary right line — 



ax + by + cz + dw = 0, alx + Vy + c'z -(- dw — 0, 

 the result of elimination, i? = 0, must be satisfied identically. This resultant 

 will be of the second degree in abed and in a'b'c'd' ; of the second, in x'y'z'w', 

 and of the 4?z — G" in xyzw. 



* N. B. — x-.y!Z2Wi is here considered as variable; ■t^iw, as fi.xed. 



