198 
IOWA ACADEMY OF SCIENCE Voi,. XXVII, 1920 
junction of the lines is similarly ineffective as regards the penta- 
hedron. 
The desirable connection would be given by a four parameter 
form which has rational lines and rational pentahedron in a do- 
main as simple as possible. 
In discussing this problem the question of a group of transfor- 
mations of the surface into itself is met at the outset. Some easily 
attainable results are : — 
1. If the group is continuous of non-multiplicative type the 
surface is ruled. 
2. If continuous of multiplicative type is is singular. A list 
can be made and the examples correlated with Schlafli’s 
classes. 
3. For the general case, since the pentahedron must transform 
into itself there must be equalities among the coefficients 
of Rodenburg or the group is the identity. In this case 
we should have the truly general form. 
The possible groups are of order 120 as in Clebsch’s diagonal 
surface, 24, 12, 6 or 2. The Clebsch surface is not contained 
in Cayley’s form nor are some of the others, the trouble being 
that Cayley’s reduction of the general quadric is a special one. 
The diagonal surface is rational in the domain (V^) for both 
lines and pentahedron. 
Since Klein ® takes small distortions of the diagonal surface 
as the general type, whereas Cayley’s form is finitely removed, 
some doubt may be expressed as to the equivalence of the forms 
thus obtained with the general one. Further his remark that a 
model should be symmetrical is unfortunate since the general 
cubic has no transformation into itself and hence it is in no way 
symmetrical. 
The problem may be attacked by assuming the pentahedron and 
a triangle which involve the adjunction of cubic and quadratic 
irrationalities. The reduction of Schlafli’s form requires ad- 
junction of the roots of a quartic and cubic. 
These adjunctions are possible by rationalizing conics and cubics. 
The detailed formulae are naturally complicated and restrictions 
arise which differ according to the group of transformations into 
itself. Finally a complete exploration of the four dimensional 
space is required to show that the forms approach every point. 
State University oe Iowa. 
6 Math. Ann. 6. 
