324 G. H. KNIBBS. 



tion of the axis of generation. This last is the characteristic 

 condition. 



If A t be two-dimensional, f(x, y) say, the x and y axes, or 

 systems of axes, straight or curvilinear, lying in a non-rotating 

 plane inclined at any angle \ir ± to with the rectilinear £-axis, then 

 to determine in cubic units the generated volume V t , it is necessary 

 only to multiply by cos to, if the generatrix be expressed in square 

 units ; consequently the position of, or any. change of position in, 

 the generating elements is immaterial. 



6. Curvature of the pri7icipal, i.e. the t-axis. — Subject to certain 

 restrictions the axis t may be a plane curve, a tortuous curve or 

 curve of double curvature, or more generally a curve of ri-ple 

 curvature. Let the radius of the osculating circle at any point 

 in the curve, first of all assumed to be plane, be denoted by p. 

 Then in order to be unequivocally specified, the abscissae of the 

 generating line or lines must, whether rectilinear or curved lie in 

 this radius, or the radius produced, or in the plane containing 

 the radius and orthogonal to the plane of the curve, the surface 

 generated by any element 8x or 8s, rectangularly distant Ap from 

 an axial-sur/ace, defined by the path of curved axis t itself, 

 moving perpendicularly to its own plane, (or, what is the same 

 thing, by the sheaf of lines drawn through the axis t, continually 

 perpendicular to the plane in which it lies) — will be greater than 

 that generated by an element of the same length in this axial- 

 surface, in the ratio 1 to (1 + Ap/p), which ratio will be a factor of 

 correction. Hence in order that the correction involved by the 

 curvature of the axis (and consequently of the axial-surface) shall 

 disappear, it is necessary that 



2(8s.Ap) = (6) 



for every value of the abscissa. That is to say the centre of 

 inertia of the system of lines must lie continually in the axial- 

 surface as defined. 



Subject to certain further restrictions the axial-surface may also 

 be oblique to the curve, that is to say in generating this surface 

 the direction of the path of every point on the curve may be 



