86 Journal of the Mitchell Society. [June 



reduced to the canonical form uvw — &i£ = 0, where u, 

 v, w, £, 77, I are linear poly momes. 



The number of independent constants in the general equa- 

 tion of the third degree is V)\ n ^ + ^ + U \ for n = 3I 



L o J 



Since the linear polynomes u, v, w, £, rj, £ contain 18 ratios of 

 coefficients and there is one other constant factor implicitly 

 contained in one of the products uvw, &?£, therefore the forms 

 uvw — £r}{, = contains 19 constants and is one into which 

 the general equation of a cubic surface may be thrown. 



It will appear later (§15) from geometrical considerations 

 that the problem to reduce the base cubic to the form 

 uvw — £tj£ — is soluble in 120 different ways. 



Notation. Consider the canonical form of the surface of 

 the third degree ace — bdf = 0, where a, b, c, d, e, f are 

 linear polynomes. By inspection it is patent that this surface 

 contains the nine lines ab, ad, af, cb, cd, cf, eb, ed, ef where 

 ab, for example, represents the line of intersection of the 

 planes a = 0, b = 0. If we suppose a = fib to be the equa- 

 tion of one of the triple tangent planes through the intersec- 

 tion of the planes a and b, the plane a = fib meets the surface 

 in the same lines in which it meets the hyperboloid 

 fi.ce — df '= 0, that is, the two lines in the plane are gener- 

 ating lines of different species, and consequently one of them 

 meets the pair of lines cd and ef, and the other of them meets 

 the pa ir of lines cf 'and ed. Let us now denote each of the remain- 

 ing eighteen lines by the three lines which it meets, the line 

 meeting ab, cd and ef being denoted by the symbol ab • cd • ef. 

 Since /a has three values, there are three lines that meet ab, 

 cd, ef Applying the same reasoning to the planes through 

 be and ca, we employ the following symbolism for the twenty- 

 seven lines ab, ad, ef (ab • cd • ef) » , (ad • cf- eb) », 



(af-cb-ed)i, (ab-cf-ed)i, (ad>cb>ef)i, (af'cd-eb)i, 

 where * = 1, 2, 3. 



