XI] 



ITS GENERAL PROPERTIES 



499 



or else by a spiral curve, somewhere within reach and recognition 

 of it. 



We come to a similar result when we consider, for instance, 

 a cyHndrical body in which forces of growth are at work tending 

 to its elongation, but these forces are unsymmetrically distributed. 

 Let the tendency to elongation along AB be of a magnitude pro- 

 portional to BB', and that along CD be of a magnitude proportional 

 to DD' ; and in each element parallel to AB and CD, let a parallel 

 force of growth, proportionately intermediate in magnitude, be at 

 work : and let EFF' be the middle line. Then at any cross- 

 section BFD, if we deduct the mean force FF', we have a certain 

 positive force at B, equal to Bb, and an equal and opposite force 

 at D, equal to Dd. But AB and CD are not separate structures. 



F 



"F 



Fig. 240. 



but are connected together, either by a solid core, or by the walls 

 of a tubular shell; and the forces which tend to separate B and 

 D are opposed, accordingly, by a tension in BD. It follows there- 

 fore, that there will be a resultant force BG, acting in a direction 

 intermediate between Bb and BD, and also a resultant, DH, 

 acting at D in an opposite direction; and accordingly, after a 

 small increment of grpwth, the growing end of the cyUnder will 

 come to lie, not in the direction" BD, but in the direction GH, 

 The problem is therefore analogous to that of a beam to which 

 we apply a bending moment; and it is plain that the unequal 

 force of growth is equivalent to a '''couple" which will impart to 

 our structure a curved form. For, if we regard the part ABDC 

 as practically rigid, and the part BB'D'D as pliable, this couple 



32—2 



