MATHEMATICAL NOTE 11 



may call it) and we can alter the direction in the tangent plane 

 of the tangent line. 



In the first place a well-known theorem, known as Meunier's 

 theorem, connects the radii of curvature of different sections 

 through the same tangent line: the radius of curvature of an 

 oblique section through a tangent line at P is equal to R cos (/> 

 where R is the radius of curvature at P of the normal section, 

 (i.e. the section containing the normal line at P as well as the 

 tangent line) and (j> is the angle between the normal section and 

 the oblique section. Thus if we know the radius of curvature of 

 the normal section through the chosen tangent line at P we im- 

 plicitly know the radius of curvature of any given oblique section 

 through it. 



In the second place if we now vary the direction of the tangent 

 line the radius of curvature of the normal section varies in a 

 manner which is well known and quite simply described. Call- 

 ing the curvature of the normal section c (where c is of course 

 equal to i2~0 it is known that c varies continuously in value be- 

 tween a maximum limit and a minimum as the tangent line is 

 rotated. It attains its maximum value twice in a complete 

 rotation of the line, the two directions corresponding to this 

 maximum being directly opposite to one another. The mini- 

 mum is attained for the two opposite directions at right angles to 

 the former. Taking the two lines thus marked out on the tan- 

 gent plane as axial lines PXi, PX^ in the plane, we can indicate 

 the direction of any other line in the tangent plane by the angle 

 6 which it makes with PXi, say. It is known that c, the curva- 

 ture at P of the normal section through this line, is given by 



c = Ci cos^ 6 -{- C2 sin^ d, 



where C: and c^ are the curvatures at P of the normal sections 

 through PXi and PX2. The values Ci and C2 are known as the 

 "principal curvatures" of the surface at the point P. In this 

 way we see that our complete knowledge concerning the curva- 

 ture of a surface at a point P is summarized in a knowledge of 

 the two principal curvatures at that point. One simple result 

 of some importance follows very easily from the equation just 

 written: if c and c' are the curvatures of two normal sections at 



