166 



Mr Bateman, The determination of curves 



dp _ 



constant, 



x =cfdoc.xsecx> 

 oc' = I dxe^~^ , 



x = jdxe Kx sec %. 



Eectdine 



dr 



cr = As, 

 dp 



When we require a special relation between the curvatures it is 

 convenient to make use 

 of the indicatrix. Now 

 it is evident that a group 

 of curves which have the 

 same tangent indicatrix 

 have also the same indi- 

 catrix for any line which 

 is fixed relative to the 

 axes of the curve. 



Draw a line OQ of 

 unit length parallel to 

 a line PIT in the rectify- 

 ing plane making a con- 

 stant angle ty with the 

 tangent. The tangent at 

 Q to this indicatrix is 

 parallel to the principal 

 normal of the curve, while 

 the binormal of the in- 

 dicatrix is parallel to the 

 rectifying line of the 

 curve. 



Let % be the angle 

 between the principal 

 normal of the indicatrix 

 and the radius of the 

 sphere. 



Then ~- — % = angle between the binormal and radius, = a — yjr, 



Li 



where a is the angle between the rectifying line and tangent of 



the curve; so that tana = -, therefore 



P 



tan x = c °t ( a ~ ^ r ) 



1 -\ — tan yjr 

 P 



P.N 



Bin: 



— tan ty 



