PROCEEDINGS OP SECTION H. 



571 



Then, as in the figure, if TCT' be the tangent to the curve at its 

 point of raaxinaum curvature C?, and the angle /^C be denoted by a, 



we have the angle ITC = 3a, and 3o = 90° — - , from which a is 



known. Also OC = 3li sin 2a, and thus the length of OC is deter- 

 mined. 



Clearlj the maximum value of a occurs when i = 0, a h then 

 ^ 30°. As the lemniscate curve may be set out through an angle of 

 45°, it is thus easily possible to set out a double lemniscate curve 

 joining two parallel lines or two Hues at any angle. 



From the triangle OIC we readily get 



01 - ZR 



sin/'l20« — ^^'j 

 2 sm - 



IC = ZR 



sin ^30^ —*.\ sin /'60° ^\\ 



IfOCO' were an ordinary simple circular curve of.radi\is R, we 

 should have 



01= R cotl , IC = ijfcosec ^ - 1 V 

 2 \ 2 J 



The following table gives the comparative values of 0/aud IC for 

 the simple circular' curve and the double lemniscate, with different 

 values of t : — 



30° 



60° 



90° 



120° 



160° 



