518 Mr Baker, On a certain system of [May 2, 



II. The five points lying on the plane at infinity given by 



y z 



where b is any one of c l} a 1} c 2 , a 2 , c, the singular tangent planes 

 which pass through this point being 



P, Pby Pbbi, Pbb 2 , Pbb^y Pbb t , 



where b 1} ..., 6 4 are the four of the set c u a 1} c,, a.,, c other than b ; 



III. The ten points given by 



os = 6j + b 2 , y-- byb.j., z = e 6iA , 



where 6 l5 6 2 are any two of c l3 a l3 c 2 , a 2 , c, the singular tangent 

 planes which pass through this point being 



-P&U -P& 2 > Pbfi 2 > PbJJi, Pbjj 5 i *■ b 3 b 5 , 



where b s , b 4 , b 5 are the three of the set c l3 a l3 c 2 , a 2 , c other than 



Still supposing X 6 = 0, X 5 = 4. the expressions for the partial 

 differential coefficients of the third order in terms of x, y, z, 

 referred to in § 1, can be taken so that the squares and the 

 products of twos of g> 222 , jp 221 , fp ai , |p m are rational integral cubic 

 polynomials in x, y, z. I have obtained the ten equations, of which 

 for instance three are 



IPIb = X 2 + A, 3 # + \ 4 # 2 + 4<x 3 + 4mcy + 4>z } 



#4 = ^o + ^y' 2 + 4>wy* - fyz, 



2^.202^^1 = \ + \y + 2\ 4 xy + 8x 2 y + 4y 2 — 4>xz. 



The first of these equations shews that all the functions f n , 

 IP221 > ^2ii, g>iu can be expressed rationally in terms of the three 

 x = @. 22 , y=@2i, ^=^222, (a remark easily generalisable to higher 

 values of p). It is known that there exist cubic surfaces touch- 

 ing the sixteen-nodal quartic surface along sextic curves ; denoting 

 the right-hand sides of these equations momentarily by X, Y, Z, 

 the equation of the quartic can be put into the form 



4>XY=Z>, 



which shews that each of the cubic surfaces X , Y touches the 

 quartic along a sextic curve lying on Z = 0. There are also 

 irrational forms for the equation of the quartic ; for instance one 

 arises from the verifiable identity g> 211 = ^ 21 ^ 222 — ^22^221 • Further, 

 in the form t 2 = % (x, y), to which the equation is reducible bi- 

 rationally, ^ (x, y) is the product of the five expressions P& . 





