334 WILSON AND MOORE. 



The conditions |ix8xh = and ynxyioxy^a are therefore equivalent as 

 was to be expected. If we use an orthogonal system of curves for 

 the parameter curves, ais = 0, and |Jix6 may be factored. If we use a 

 minimum system, an = 0-22 = 0, and JxxS reduces toynxyaaexceptfora 

 factor. In general J^xS may be factored in 00 ^ ways of which one simple 

 case is, 



2at|ix5 = ( ai2 -^ — ^^ — yi2 ) X (rt22yii — «ny22). 



\ Oil + ('22 I 



The vertex of Cone II is located at the point, 



(lJLx5).(ULx5xh) 



which may be expressed in terms of the y's if desired. 



The invariant [|ix6]'- which is proportional to the square of the area 

 of the indicatrix is except for a factor Levi's invariant A4 . The 

 invariant 



'J>2s = yii><yi2 • yi2xy22 — , [y22xyn]" (104) 



4 



is, except for a factor, Levi's invariant A3 . We have geometric inter- 

 pretations for all the invariants except <J>2s . If we write 



*2s/a2 = [H'=<5]2 _ [5xh]2 - [hxfJLp, (1040 



we have KH-xS] interpretable as the area of the triangle of which the 

 conjugate radii jx and 5 are sides; M^^ht] as the area of the triangle of 

 which 8 and h or 6 and a are sides; |[hx|x] as the area of the triangle 

 of which h and jJ- are sides. As [p.x6]2 is itself an invariant [H^-xS]'- — 

 ^2s/ct^ is an invariant and is equal to four times the sum of the squares 

 of the areas of the triangles on 8 and h and on h and |a. 



We can therefore set up the following list of five scalar invariants,^^ 



h^ $s/a = G, [|J^x8p, [|jix8]2 _ $2s/V-, [|ix8xh]2. ■ • 



39 To aid the reader to make the comparison between our notation and Levi's 

 we give the following table of equivalents for his symbols / and /. 

 /loio = Oil, /looi = fti2, •'0101 = O22, 



«/2020 = yil^, J WW = yi2', </0202 = Yl'^i 



Jioo2 = yii*ya2, •/2021 = yii«yia, Jom = y22»yi2- 



