300 G. H. KNIBBS. 



Similarly a sphere built up of elements differing in intensity in 

 three directions, but constant in each direction will be as to its 

 intensity quantum, an ellipsoid, the intensity volume of which is 

 % 7raj3yr s , if the axes be perpendicular to each other. 1 



If a line r be rotated about one end, the path of its term is of 

 course the circumference of the circle, formed by the path of the 

 line itself ; if in rotating, the radius be increased or diminished 

 proportionally to the arc through which it is turned, the term 

 traces out the spiral of Archimedes, 2 while if in rotating, it 

 lengthens or diminishes as expressed by the equation r = a#, the 

 path of the term is an equiangular or logarithmic spiral. 3 This 

 generation is continuous. 



In Fig. 26, suppose a line r, viz. RS, perpendicular to the axis 

 ZO, to increase its length, according to the law implied by the 

 equation r= V(az-z 2 ), z being the distance ZR; its term S will 

 be a spiral on a spherical surface, if it revolve with infinite velocity 

 while it moves along ZO with finite velocity, it may be considered 

 to trace out the spherical surface itself. 4 The surface generated 

 by the line RS will be the helical surface represented near Z in 

 the figure. A more complex surface will be generated, if RS be 

 a curve, changing its parameter as it moves along z, or changing 

 according to some more complex function of z; and if the z axis 

 be curved, the complexity will be still further increased. 



Thus by the most simple operations very complex geometrical 

 figures 5 may be generated. On the other hand apparently complex 

 motions may develope simple figures. For example the terms 



1 Many formulae become obvious from this point of view, e.g. volume of 

 an ellipsoid is = -§ ^abc. 



2 r=ad. 



3 This may be traced by attaching a thread to a point on the surface 

 of a right circular cone, whose axis is perpendicular to the plane on which 

 the vertex of the cone lies. Draw the thread tight and wind it round the 

 cone: if it does not slip the terminal P of the thread, kept on the plane, 

 will trace the curve. 



4 More strictly a spiral line, whose winding on the sphere is only 

 separated by an infinitesimal distance, a being finite. In respect to the 

 spherical surface this generation is not continuous. 



5 Or figures that may be regarded as representing them. 



