322 G. H. KNIBBS. 



which therefore is the factor for converting the oblique into square 

 units. So also, if A t =f{x, y), and the angle between these axes 

 be ^7r ± to', and between the x, y plane and the t axis be \ir ± co, 

 the factor to reduce the parallelepipedic to right-cubic units will 

 be cos co cos to'; and similarly if A t =f(x, y, z, w, etc.), the inclin- 

 ations being \ir ± co, co', co", co"' etc., one of these being in relation 

 to the axis t, the factor of reduction k for the w-ply oblique units 



will be 



& = cos co. cos co'. cos co". etc (3). 



It is here assumed that the axes do not rotate about the axis t. 



4. Generating junction with curvilinear axes. — If the axis t be 

 rectilinear, and x either rectilinear or curvilinear in a plane 

 orthogonal with t, the one-dimensional generatrix is subject to no 

 restriction except linear conformity to (1). Thus it may rotate 

 about the £-axis, as is evident from the consideration that the 

 element of surface 8V t = 8A t St is not affected by the direction of 

 motion in relation to a line coaxial with that axis and lying in 

 the plane containing the element; since the generated helicoidal 

 element of surface is diminished in rectangular width, in exactly 

 the same ratio as it is increased in length by the rotation. 



An identical consideration indicates that the element of volume 

 8V t generated by the motion of the surface-element 8A t 8t is not 

 affected by rotation about the £-axis, whether the units of surface 

 be square, oblique, or are curvilinearly orthogonal or oblique, 

 provided only that they remain constant in kind. This is clearly 

 quite general, so that we may substitute 'w-dimensional homaloidal 

 element' for 'element of volume,' and \n - l)-dimensional homa- 

 loidal element' for 'surface-element,' in this last theorem. 



5. Rotation of generatrix obliquely about the t-axis. — If the 

 generating element lie in a plane not orthogonal with the £-axis, 

 and not rotating about that axis, then an element of surface 

 generated by linear element, will, when reduced to square units, 

 be 8 V t = kA t 8t, in which k is constant only for parallel directions 

 in the plane, or rather for two series of parallel directions, whose 

 axes of symmetry are the directions which give the maximum 



