IX] OF INTRACELLULAR SPICULES 441 



For obviously, if they lie in a line perpendicular to the long axis 

 of the ellipsoid, the geodetic which connects them is a circle, also 

 perpendicular to that axis ; and if they lie in a line parallel to 

 the axis, their geodetic is a portion of that ellipse about which 

 the whole figure is a solid of revolution. But if our two points 

 lie, relatively to one another, in any other direction, then their 

 geodetic is part of a spiral curve in space, winding over the surface 

 of the ellipsoid. 



To say, as we have done, that the geodetic is the shortest line 

 between two points upon the surface, is as much as to say that 

 it is a projection of some particular straight line upon the surface 

 in question ; and it follows that, if any linear body be confined 

 to that surface, while retaining a tendency to grow by successive 

 increments always (save only for its confinement to that surface) 

 in a straight line, the resultant form which it will assume will be 

 that of a geodetic. In mathematical language, it is a property 

 of a geodetic that the plane of any two consecutive elements is 

 a plane perpendicular to that in which the geodetic lies; or, in 

 simpler words, any two consecutive elements lie in a straight line 

 in the plane of the surface, and only diverge from a straight line 

 in space by the actual curvature of the surface to which they are 

 restrained. 



Let us now imagine a spicule, whose natural tendency is to 

 grow into a straight linear element, either by reason of its own 

 molecular anisotropy, or because it is deposited about a thread- 

 hke axis ; and let us suppose that it is confined either within a 

 cell- wall or in adhesion thereto ; it at once follows that its line 

 of growth will be simply a geodetic to the surface of the cell. 

 And if the cell be an imperfect sphere, or a more or less regular 

 elhpsoid, the spicule will tend to grow into one or other of three 

 forms : either a plane curve of circular arc ; or, more commonly, 

 a plane curve which is a portion of an ellipse ; or, most commonly 

 of all. a curve which is a portion of a spiral in space. In the 

 latter case, the number of turns of the spiral will depend, not only 

 on the length of the spicule, but on the relative dimensions of 

 the ellipsoidal cell, as well as upon the angle by which the spicule 

 is inclined to the ellipsoid axes ; but a very common case will 

 probably be that in which the spicule looks at first sight to be 



