742 A PARENTHETIC NOTE [ch. 



are all of frequent occurrence, and are illustrated in the local 

 thickenings of the wall of the cyHndrical cells or vessels of plants. 



The spiral, or rather helical, geodesic is particularly common in 

 cylindrical structures, and is beautifully shewn for instance in the 

 spiral coil which stiffens the tracheal tubes of an insect, or the 

 so-called tracheides of a woody stem. A Hke phenomenon is 

 often witnessed in the splitting of a glass tube. If a crack appear 

 in a test-tube it has a tendency to be prolonged in its own direction, 

 and the more isotropic be the glass the more evenly will the split 

 tend to follow the straight course in which it began. As a result, 

 the crack often continues till our test-tube is split into a continuous 

 spiral ribbon. 



One may stretch a tape along a cylinder, but it no sooner swerves 

 to one side than it begins to wind itself around; it is tracing its 

 geodesic*. 



In a circular cone, the spiral geodesic falls into closer and closer 

 coils as the cone narrows, till it comes to the end, and then it winds 

 back the same way; and a beautiful geodesic of this kind is 

 exemplified in the sutural line of a spiral shell, such as Turritella, 

 or in the striations which run parallel with the spiral suture. On 

 a prolate spheroid, the coils of a spiral geodesic come closer together 

 as they approach the ends of the long axis of the ellipse, and wind 

 back and forward from one pole to the other "j". We have a case of 

 this kind in an Equisetum-sipoTe, when the integument splits into the 

 spiral "elaters," though the spire is not long enough to shew all its 

 geodesic features in detail. 



We begin to see that our first definition of a geodesic requires to 

 be modified; for it is only subject to conditions that it is ''the 

 shortest distance between two points on the surface of the solid," 

 and one of the commonest of these restricting conditions is that 

 our geodesic may be constrained to go twice, or many times, round 

 the surface on its way. In short, we may re-define a geodesic, as 

 a curve drawn upon a surface such that, if we take any two adjacent 



* It is not that the geodesic is rectified into a straight line; but that a straight 

 line (the midline of the tape or ribbon)^ is converted into the geodesic on the given 

 surface. 



t In all these cases, r cos a = a constant, where r is the radius of the circular 

 section, and a the angle at which it is crossed by the geodesic. And the constant 

 is measured by the smallest circle which the geodesic can reach. 



