Roberts — On Binary Cubics and their Associated Forms. 529 



Now It; = A,/p./, 



therefore ^R,;R^ = 2 Aj^A^^jy^ + A/^;^ , 



or 3(i2vy-^. = 2(Av)A,(iJV) + (^V)V., 



which expresses {R-^yR^ in terms of Clebsch's forms. 

 In the same way we can represent the form 



2(7rv)(Av)Vx+(vA)V.. 



XIII. 



"We now turn to the invariants, of which there are seven, two 

 being combinative. 



Let us seek the condition that the point may lie in the plane <p, 

 and we know, by what we have proved in I., that when this condition 

 is fulfilled, 0' lies in the plane /. 



Expressing the condition that the point a-^, - «,, ai, -«o niay lie in 

 the plane ao^+ 3ai F+ ^a^Z+ a.,W=0, 



we find {aaf-0, or /= 0. 



Hence J vanishes when lies in the plane of </> ; but in this case the 

 line/, <^, becomes identical with its corresponding line, and this rela- 

 tion is evidently combinative, since it depends only on the line /, ^. 



= is the condition that must be satisfied in order that/+i-^ 

 may become a perfect cube ; hence, when it vanishes it will be possible 

 to draw an osculating plane through the line/, <^, and for the same 

 reason as-before this relation is combinative. It is easy to see also 

 that when O vanishes that the Kne corresponding to / 0, or the line 

 0' meets the curve. 



Now, Clebsch has shown analytically that 2Q, = {p-), and we can 

 show geometrically that when O vanishes, or that when the line 00' 

 meets the curve in a point 0", that the points ji; and tt coincide. The 

 point p is the point in which a plane containing 0' and the chord 

 through meets the cuj-ve again ; but this plane contains the line 

 0', which, by hypothesis, meets the curve in 0" ; hence, when = 0, 

 p coincides with 0", and in like manner tt is shown to coincide with 

 0", and therefore with p. 



The two discriminants have been discussed before as the conditions 

 that the points and 0' should lie on the developable formed by 

 tangent lines to the curve, respectively. 



We now come to the invariants 



% = {aKy = {®^r, 



Now it is easy to see that 2 is the condition that should lie in 

 the plane K, or that the line /, K should coincide with its own con- 

 jugate. *S', in like manner, is the condition that 0' should lie in the 

 plane Q. 



