satisfying given conditions. 165 



Now the quantity u is at our disposal, accordingly if we choose u 

 so that 



F {pu, <tu, udx, ady, adz] = 0, 



the new curve will possess the property 



F{p, a' y dx',dy', dz'} = 0, 



thus for a curve of constant torsion 



1 



u = — . 



O" 



Of the curves in which there is a special relation between the 

 curvatures, Bertrand's curves form an interesting class: they 

 possess the characteristic property that their principal normals 

 are also the principal normals of another curve. The relation 

 in this case is of the form 



a b 

 - + - = c, 

 P <* 



this includes helices ( - = constant J , curves of constant curvature, 



and curves of constant torsion. 



Curves of constant torsion have been much studied during 

 recent years: in 1884 Darboux proposed the problem of finding 

 the algebraic curves of constant torsion, several attempts have 

 been made and some particular curves have been obtained*. 



Returning to equations (3) it is evident that we shall obtain 

 the same results whichever curve of the group we start with ; 

 now as there will always be a number of curves of the group which 

 can be drawn on a sphere, suppose we choose one of these as our 

 initial curve. 



If x be the angle between the radius of the sphere and the 

 principal normal of the curve, we have the formulae 



P = a cos X , o- = ^ . 



The following are the forms for the equations of the derived curve, 

 corresponding to different properties : 



, , [dx , 



a=C, x=cJ Ts d X , 



p — c, x = c /sec % . dx. 



Curve on a sphere having same indicatrix 



x = cjdx {a + b tan x}> 



* Cf. Darboux, Theorie generale des surfaces, Vol. iv. p. 429. 



