334 WILSON AND MOORE. 



The conditions |i.x8xh = and yiixyi2xy22 are therefore equivalent as 

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

 the parameter curves, an = 0, and p.x6 may be factored. If we use a 

 minimum system, an = 0^2 = 0, and p.x8 reduces to ynxy22exceptfor a 

 factor. In general p-xS may be factored in 00 3 ways of which one simple 

 case is, 



2«ii^|ix6 = ( an ^^ — '^- — yi2 ) X (ff22yu — oiiy22). 



\ (In + a«2 / 



The vertex of Cone II is located at the point, 



(|xx6).(|jLx5xh) 

 (H^xbxh)- 



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



The invariant [|ix5]2 which is proportional to the square of the area 

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

 invariant 



$2s = yiixyi2*yi2xy22 - , [y22xyn]^ (104) 



4 



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

 pretations for all the invariants except $2s • If we write 



^2s/a2 = [|xx5]2 - [8xh]'-^ - [hx|Ji]2, (104') 



we have MH'x5] interpretable as the area of the triangle of which the 

 conjugate radii H- and 8 are sides; M^^h] as the area of the triangle of 

 which 8 and h or 8 and a are sides; |lhx|i.] as the area of the triangle 

 of which h and p. are sides. As [K-x8]2 is itself an invariant [|J-x8]2 — 

 ^2s/tt" is an in\ariant 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 p.. 

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



h?, *s/a = G, [Jix8]2, [|xx8]2 - $2s/a-, [fix8xh]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 /. 



/lOlO = On, ■'1001 = Otl2, /oiOl = fl22, 



./^2020 = yil^, JnU = yi2", ^^0202 = y22^, 



«/2002 = yil*y22, «^2021 = yil«yi3, JoiU = y22»yi2- 



