678 MOORE. 



But since Mi-r and M^-r are vectors of constant length and lie in 

 perpendicular planes their cross product has constant magnitude. 

 Therefore the scalar first torsion of the path curves is constant. 



From the above expression for T we see that if m\ = or im = 

 or »?i = =t ???2 the torsion vanishes, that is the plane txc is independent 

 of the point on the curve. Therefore in the group of infijiitesimal 

 transformations ichich leave the same pair of perpendicular planes Mi 

 and jM2 invariant there are four transformations whose path curves are 

 plane curves. If vii = the motion reduces to a rotation parallel to 

 the plane Mo and the path curve is the circle whose plane is parallel 

 to Ml and whose center is on Mi. Likewise if m^ = the path curve 

 is a circle in a plane parallel to J/o and whose center lies in Mi. If 

 nil = =*= w'2 equation (42) becomes 



C = mi'[Mi-iMi-r) + M^'iM.-r)] (^^ 



But since Mi-{Mi-r) and Mo- (Mo-r) are the projections of the vector 

 r on the planes Mi and J/2 respectively and these planes are perpendicu- 

 lar the expression in the brackets is evidently equal to r. Hence in 

 this case 



From (41) we see that in this case 



(!;=..(..). 



Then 



r 



and hence 



C = 



r-r 



C-C= — . 



r-r 



That is the scalar of the curvature of the path curve is the reciprocal 

 of tlie length of r. Then in this case the path curves are circles with 

 center at the origin. The plane of the circle is rx(J/i-;- -f- J/g-r). 

 This plane changes only in magnitude if we change the length of r. 

 Hence the plane is left invariant by the transformation dr = mi(Mi + 

 Mo) - r dt. That is through each point in space passes one plane left 

 invariant by this transformation. From (.39) the directions of the 



