676 MOORE. 



IV. 



S. Surfaces in 4-space left invariant by all the rotations 

 having the same two fixed planes. All the rotations represented 

 by the equation 



(39) r'= /• + M-r dt = r + (lUiMi + imMo^-rdt 



where J/i and Mi are unit planes and iiii and ?»2 are allowed to vary 

 form a group since each transformation of the set leaves 3/i and J/2 

 invariant and consequently the product of two of them will leave these 

 planes invariant also. The direction which a point will move by 



(39) with fixed values for mi and mo is 



dr 



(40) J = (wiJ/i + m2Mi)-r = M-r. 



If mi and m^ are allowed to vary it is seen at once that all the directions 

 which a given point can take lie in a plane since they are linear func- 

 tions of the two vectors Mi-r and M^-r. It is seen also from this 

 equation that the ratio nii: vio, is all that need be considered since their 



actual values are necessary for determining the magnitude of 77 and 



not its direction. If we give ?»i and m^ definite values (40) will be 

 the (Hfferential equation of the path curve described by the end of the 

 vector /• by this particular rotation. The unit tangent to this curve is 



r = -=i/.r- 

 ds ds 



Since the magnitude of r is unity we have 



dr dr , (dt^^ 



= [m{'{Myry + m,KMo-ry]\^-y 



Hence 



(41) i^jj = miKMi-rf + mi^{Mo'r)\ 



As the transformation is a rotation r is a vector of constant length 

 and also the projections on the fixed planes 3/i and M2 are also the 



