688 MOORE. 



will correspond to four different planes of directions through the point. 

 These are given by the linear relations 



a\ -\- b/jL + cp = 0, 



o\ -]- b/jL — cv = 0, 



a\ — biJi -\- cv = 0, 



a\ — b/jL — cv = 0. 



By substituting these four relations in (53) it is seen that we obtain 

 the same conic. The same point in the curvature triangle correspond 

 to the perpendicular direction for each relation. If ?/h = or ???2 = 

 or mz = 0, the corresponding points in the triangle are on the sides of 

 it. The rotations in this case are of the four dimensional type previ- 

 ously discussed. 



The unit osculating plane is again the cross product of the unit 

 tangent and the unit curvature. The unit curvature is 



mmi • (Ml • r) + mm2 • (^2 • r) + mg'Ms ■ {JSh ■ r) 



C = 



mi%Mvry- + m2'(M2-ry + im'iMz-ry 



The rate of change of the osculating plane (txc) with respect to the 

 arc is again the first torsion. 



'[Mr(Mrr)]}(~y ^ 



,ds/ VC-C 



Mi- (Mi-r) is the projection of r on Mi and Mi-\Mi- (il/j- r)] is a vector 

 of equal length in Mi and perpendicular to Mi- (Mi-r) and is there- 

 fore equal to (Mi-r). Hence we can write for the torsion 



T = (■ZmiMi-r)x{lmi'{Mi- r) \ (~^' ^ 



ds) VC-C 

 = miVhimi- — ?»2') (il/i-r}x(il/2-r) + mimz{mi^ — mz-) (il/i-r)x(il/3-r) 



ldt\^ 1 

 + mimzimi — mz-) (M2-r)x{Mz-r) - 



,ds/ VC-C. 



For given values of mu 7»2, viz the magnitude of this \ector is constant. 

 The path curves are then curves for which the rate of change of the 

 unit osculating plane with respect to the arc is a vector of constant 



