MATHEMATICS: L. P. EISENHART 293 
we have stated to be characteristic of a transformation K. But as we 
have just shown, the surfaces 2 and 2i, being polar transforms of two 
surfaces in the relation of a transformation K, are themselves in the 
relation of a transformation Q. Hence we have proved the above 
theorem and also the following: 
When two surfaces S and Si are in the relation of a transformation K, 
their polar transforms are in the relation of a transformation 12; and con- 
versely. 
Because of the dual relation between these two types of transforma- 
tions, we are enabled to add to Theorems 4 and 6 of memoir Mi the 
dual of the last part of the theorem of §4 of memoir M2, and thus 
have the following theorem of permutability of the transformations K: 
If Si and S2 are two surfaces arising from S by transformations K, there 
can he found by quadratures an infinity of surfaces S\ each of which is 
in the relation of transformations K with both Si and 6*2. // M, Mi and 
M2 denote corresponding points on S, Si and S2, the corresponding points 
M' on the surfaces S' lie on a line through M and in the plane it of the 
points M, Ml, M2. The corresponding tangent planes to the surfaces 
S' envelope a quadric cone to which are tangent the tangent planes to 5, ^i, 
^2 at M, Ml, M2. Moreover, the plane it touches its envelope at the point 
of intersection P of the lines MM' and M1M2; and the parametric curves 
on the envelope form a conjugate system whose tangents are harmonic to the 
lines MM' and M1M2, and contain the focal points of the lines MMi, 
MM 2, M'Mi, M'M2. An analogous theorem of permutability of trans- 
formations follows from the above in accordance with the principle 
of duality. 
The relation between the two types of transformations is likewise 
helpful in interpreting the significance of certain evident forms of the 
transforming functions di and ivi. Thus we have shown {Mi, p. 401) 
that the necessary and sufficient condition that for two surfaces 6* and 
Si in the relation of a transformation K the corresponding tangent planes 
be parallel is that di be constant. In this case S and Si are associate 
surfaces, that is not only are the tangent planes parallel but also to 
asymptotic lines on either surface corresponds a conjugate system on 
the other. Moreover, any two associate surfaces are in this special 
kind of relation of a transformation K. Let S and Si be two associate 
surfaces and apply to them the polar transformation with respect to 
the quadric (5), where a, b, c are different from zero. Since correspond- 
ing tangent planes to S and Si meet in a line in the plane at infinity, 
the lines joining corresponding points on 2 and Zi meet in a point — 
the pole of the plane at infinity with respect to the quadric. From (7) 
