GENEEATION OF FIGURES ON STRAIGHT OR TORTUOUS AXES. 323 



and minimum values of k. If this plane make an angle of \tt ± o> 

 with the £-axis, then k, for any line therein making an angle f 

 with that direction which is perpendicular to t, 1 will be 



&=V(l-sin 2 f sin 2 w) (4) 



from which it is evident that, in general, the generating element 

 is not free to rotate about the axis t, since k varies with £ unless 

 sin 2 co = 0, that is unless the axis of the generatrix-plane, or the 

 generatrix system of axes, is orthogonal with t. It is therefore 

 essential that the mean value of k, k say, should be constant for 

 every value of t, that is to say we must have 



k = — *2(k8A t )= constant (5) 



This condition will be satisfied if the projections of the linear 

 figure or figures A t for successive values of t, on a plane perpen- 

 dicular to the t axis, are either homothetic, or are similar and 

 similarly oriented with respect to the axes of the plane of projec- 

 tion. If the generating function represent a circle or a series of 

 circles, its or their motion is unrestricted, because they will project 

 as ellipses with parallel axes in the ratio cos w : 1; but if it repre- 

 sent the perimeter of a polygon or indeed any other figure than a 

 circle — or a series of such — it or they must not change its or their 

 orientation, with respect to the plane of the axial-inclination w, 

 though otherwise unrestricted as to movement. Rotation in the 

 plane of the generatrix involves in general therefore an opposite 

 rotation of equal amount, of the rotated lines or figures about 

 their own axes, in order to satisfy condition (5). 



The necessity for imposing this condition is a consequence of 

 the fact that motion, in the plane of a generatrix, oblique to the 

 axis of generation, (t), is essentially an unequal variation in the 

 rate of motion in the latter, of the several elements of the gener- 

 ating function : the condition that the projections shall be either 

 homothetic, or similar and similarly oriented, involves essential 

 uniformity in the motion, of the generative elements, in the direc- 



1 That is with the line of intersection with a plane perpendicular to 

 the t axis. 



