326 G. H. KNIBBS. 



variable. The projection of the X axis (PX say), on the plane 

 perpendicular to the radius of the osculating circle, i.e. the Y axis 

 (PY say), makes an angle \tt ± u>" with the intersection of the 

 plane of osculation therewith, the angle to'" being given by the 

 equation tan w - = tan w cos M > ^ 



The factors for reducing an element 8s of the generatrix at right 

 angles to the axis PY is cos a/"; or if the angle made by the 

 direction of this element with the Y axis be denoted by f3, the 

 reducing factor (k') is the sine of the angle of inclination (x) 

 between the direction of 8s and that of the tangent to the curved 

 axis t at the abscissa-plane in which 8s lies. This is given by the 

 expression 1 



A;' = sin x = cos P V(l - tan 2 /3 cos 2 to'") (9) 



hence the mean value of k\ k' say, for each successive position of 

 the generatrix must be constant, and the condition expressed in 

 equation (5) therefore hold good. 2 The necessity for simultane" 

 ously satisfying conditions (6) and (5a) say, indicates how greatly 

 the oblique relation, of the axial-surface with the plane of the 

 £-axis, complicates the issue for a 1-dimensional generatrix. It is 

 otherwise when the function is 2-dimensional, that is when A t = 

 f(x.y), a case we now proceed to consider. 



First suppose the axes of the plane surface A t , continually per- 

 pendicular to the plane of the £-axis, to be rectangular, one axis 

 being the radius of the osculating circle: then (6) is the only con- 

 dition to be satisfied, 8s becoming 8x.8y. If the plane of the 

 generatrix has the oblique relation previously defined, the generated 

 volume has simply to be reduced by multiplying by cos to, and a 

 condition similar to (6) has alone to be satisfied, Ao being inter- 

 preted as in the preceding case, that is to say it is to be always 

 measured in a line parallel to the plane of the curved £-axis. The 

 element 8A t of the generatrix being however 8x.8y. cos to", see 

 formula (7), the value of the generatrix is not simply f(x. y) 



1 It may be noticed that if a/" = 0, k is unity, as it should be, for all 

 values of /3. 



2 We may call this condition (5a). 



