692 MOORE. 



relation in v, r, w. This substituted in (60) shows that the surface A' 

 is also developable. This surface differs from that studied in 4-space 

 since for this one the indicatrix is a true ellipse and not a linear seg- 

 ment counted twice. The plane of the indicatrLx does not pass through 

 the surface point. In particular the linear relation m i = will lead 

 to a geodesic surface all of whose geodesies are curves lying in a 4-space. 

 This will cut iv in a geodesic. Hence passing through each point of 

 K pass three geodesies which lie in a 4-space. 



For each point on V^ there is a curvature triangle. The locus of 

 these triangles consists of the planes cutting three fixed circles, one 

 lying in each of the planes Mi, Mo, M3. 



We have found plane curves (circle) and curves lying in a 4-space 

 which are left invariant, that is, path curves. It is evident that if a 

 space curve is left invariant the space in which it lies must be left 

 invariant. We saw that no 3-spaces were left invariant hence there 

 are no 3-space path curves. Also we saw that the only 4-spaces left 

 invariant were xl/ixJ/2, il/ixJ/3, MoxMs hence the 4-space path curves 

 mentioned above are the only ones that exist. 



10. Rotations in space of 2p dimensions which leave the 

 same set of p mutually perpendicular planes invariant. Hav- 

 ing considered the case of four and six dimensions we can now easily 

 generalize the results for space of 2jj dimensions. Let the rotation 

 be expressed in terms of the unit invariant planes 



p 



(61) r'= r-^-lmiMi-rdi 



1 



or r = — = ZviiMi-r—. 



as 1 as 



The OOP transformations obtained if ?», vary, form a group and the 

 different directions which a point can take by the various transforma- 

 tions of the group lie in a linear 2>space. The curvature of the path 

 curves at the point P is given by the formula 



(62) ^ j 27niWr(J/i-r) 



Zm^iMi-rY- ' 



For given values of m , the length of this vector curvature is seen to be 

 independent of the position which r can take by the given rotation. 

 That is the path curves are curves of constant curvature. These 

 curvature vectors generate a p-space which is completely perpendicular 



