6 MeadoWCKOI T, 'rrcai})!cnt of Geodesic Torston. 



however, although necessarily somewhat longer, is very 

 elegant and affords an interesting application of the use 

 of Serret's formula;. 



Let P {x, y, s) and Q {x+di',r + dj\ :: + ds) be two 

 neighbouring points on a line of curvature. Let (/, ;//, «), 

 (A ^) 0> (^' l^> ^)> (^'' ^> ^) ^^ ^^^ direction cosines of the 

 tangent, principal normal, binormal and normal to the 

 surface at P respectively, and let the same quantities 

 with increments be the direction-cosines of the corre- 

 sponding lines at Q. Now the normals at P and C^ 

 intersect since the curve is a line of curvature. 



.'. X - f^ip = X + dx - {/.i + dfi)p, 



with two similar equations, p being the radius of curvature 



of the curve. 



du dv div I 



' ' dx~ dy~ dz~ p ^ '' 



"^ow cos\ = up + vq-\- It' y. 

 .'. - sin xdx = ("dp + vdq + 7C'dr) + {pdu + qdv + rdw) 

 = u{Xdri - hid) + viiidi) - f>/<iO) + 7t'(t'd}] - ndd) + (pdu + qdv + rdiv^,, 



by Serret's formulae, 



= dr\(u\ + z;// + «'»') - dQ{id + vm + ivn) + -{pdx + qdy + rdz), 



from (i.j, 



= dx] , cos( - + X )) since ul-\- vm + 7^7/ and pdx + qdy+rdz 



are both zero. .'.dx = dr], which is Lancret's theorem. 



in. To FIND Expressions for the Curvature 

 AND Torsion of a Line of Curvature. 



An expression for the curvature of a line of curvature 

 may be obtained by using a method given by Frost* for 

 finding the position of the osculating plane. Let PR be 



* Fro.st's "Solid Geometry," third edition, p. 285. 



