848 MECHANICS OF GROWTH. 



case where the curvature is greatest, that is where its radius is smallest; above and 

 below this the curvature is less, and therefore the radii are larger. 



It appears also that from the commencement to the termination of the process the 

 form of the curvature is always altering, the maximum of curvature being attained by 

 parts which were at first not curved at all or only slightly so, and parts which were pre- 

 viously strongly curved becoming straight. 



The following paragraphs serve to explain the foregoing. We assume, for the sake 

 of simplicity (excluding other possible cases), that the horizontally-placed shoot is 

 rooted, or that its base which has ceased to grow (and which can absorb water) is 

 fixed, whilst its apex can move freely. To make it more intelligible, let us consider 

 the whole growing region, the region, that is, which takes part in the upward curva- 

 ture, as divided into three parts, an apical, a middle, and a basal portion, which we 

 may assume to be of equal lengths. 



Since the form of the curvature of the whole curved portion is determined by the 

 degree of curvature of each transverse zone, it is essential to know upon what 

 conditions the curvature of each zone depends. The following are the determining 

 conditions : — 



1. The rate of growth. 



2. The thickness. 



3. The deviation from the vertical. 



4. The time during which any zone lies in any given direction inclined to 



the vertical. 



5. The persistent eifect. 



6. The rigidity and elasticity. 



The curvature is greater, ceteris paribus, in any given short period of time, the more 

 rapid the rate of growth in length, and the more nearly the deviation from the vertical 

 approaches the horizontal; on the other hand, geotropism is slower the thicker the 

 curving region is. Further, the curvature increases, that is the radius of curvature 

 becomes smaller, the longer the time during which the curving region is inclined at 

 an angle to the vertical. Moreover each transverse zone tends, according to what 

 was said above, to curve more strongly than is due to its inclination to the vertical 

 and to the length of time during which it is in that position ; that is, each transverse 

 zone which has been exposed for a certain time to the action of geotropism under- 

 goes in consequence of its persistent effect a subsequent curvature, which is in excess 

 of that produced by the other conditions. Finally, as regards rigidity and elasticity, 

 it is clear that each transverse zone of a shoot lying horizontally must, by reason 

 of the flexibility of the shoot, tend to bend downwards, that is, in opposition to the 

 geotropic curvature, and this tendency will be greater the greater the weight which 

 the shoot has to bear at its growing end and the more distant the section is from 

 that end. It must be further borne in mind that the flexibility alters with age and 

 that it diminishes as the thickness increases. 



If the growing region of a horizontally -placed internode or stem were of the same 

 thickness throughout, and if the rate of growth of all transverse zones were uniform 

 and the flexibility so slight that it might be neglected (as is the case in short, thick 

 stems), the curvature, at its first appearance, would have the form of a segment of a 

 circle of large radius. Of these conditions, however, one at least, viz. the uniform 

 rate of growth of all transverse zones, is never fulfilled, and since the region of most 

 rapid growth is also that of greatest curvature, it is impossible that the curvature should 

 be, even at the commencement, a segment of a circle. 



Taking now the usual case, in which we have a shoot growing at its apex, of a conical 

 form, growth being more active near the apex than near the base, the curvature re- 

 sulting from a horizontal position will first be manifested by the apical portion, for its 

 growth is the most rapid, it is the thinnest portion, and it has the least weight to raise ; 

 at a later period a less sharp curvature of the middle portion will be observed, and 



