168 Mr Bateman, The determination of curves 



gives a curve for which 



and when -f = ^ , «' =/P(tan %)O dy o - 2/o^o) 

 gives a curve for which 



Consider two such curves obtained from a given curve on the 

 sphere, the tangent at any point on one is parallel to the binomial 

 at the corresponding point of the other, whilst the principal 

 normals are parallel. 



Suppose now that we divide the line joining corresponding 

 points on the two curves in a constant ratio cos yjr : sin yjr, then 

 the co-ordinates of this point will be 



°° = cos t/t + sin Tlr -^( tan X> ^° ds ° cos ^ + ( z ° d y° ~ y« dz °) sin ^' 

 accordingly it will describe a curve for which 



(cosi/r sini/r) „ 

 \ P <? J 



COSi/r sim/r ( 



cos yjr sin t/t 



- = constant ; 



9 °" 



also the tangent at any point of this curve will make angles 



IT 



•\|r and -= — i/r with the tangents at the corresponding points of the 



other two curves, and the principal normals of the three curves 

 will be parallel. 



If we form a surface by dividing the line joining any two 

 points on the two curves in the ratio cos yjr : sin -v/r, the previous 

 curve will be a geodesic on this surface. 



Let P and Q be the points on the curves, R the corresponding 

 point on the surface. 



Now if we keep Q fixed the 

 tangent plane at R will be parallel 

 to the tangent at P, for a similar 

 reason it is also parallel to the tan- 

 gent at Q; therefore when we keep 

 Q fixed the tangent planes at two 

 consecutive points R, R' will inter- 

 sect in a line parallel to the tangent 

 at Q; accordingly the directions of 

 the tangents at P and Q are conju- 

 gate directions of the surface at R. ^~~~~o~ 



