76 RELATIONSHIPS OF SYMMETRY 



divergences run successively through all possible values between 180 and this 

 limit. Hence the members of the Schimper-Braun series of divergences 



1 1 2 3 _5^ _8_ 13 

 2' ' 5' ' 13* 2l' 34" 



have only a special significance in so far as they, being the successive approximate 



values of the continued fraction 



1 



2 + 1 



1 + 1 



1 + . . . 



through which as is well known the above-mentioned limiting value can be ex- 

 hibited, express approximately by the smallest figures the actual divergences. 



What has been shown in this example for the positions in the chief series 

 holds in general also for every other spiral system. A longitudinal pressure 

 produces always a gradual approximation to a certain limiting value, and that 

 a longitudinal pull must bring about displacements in reverse succession goes 

 without saying. 



We have hitherto assumed for simplicity's sake that the lateral organs are 

 constant in diameter and that the circumference of the mother-organ only is 

 variable. But such a supposition represents no real case ; the shoots grow always 

 so strongly that the mutual distances in the longitudinal direction become also 

 gradually greater. Whilst then the angle of the given span in consequence of the 

 predominating growth in thickness of the stem opens more and more, the two 

 rafters lengthen at the same time. Instead of a sinking of the apex as has been 

 above depicted there is, as a matter of fact, a gradual rising of the gable. The 

 lateral oscillations however will attain in this case also the same amount, as they 

 depend only upon the mutual relationships of the length of the rafters. 



The circular cross-section hitherto supposed for the lateral organs is almost 

 completely realized in many cases in nature, especially in the region of the flower, 

 but in numerous other instances in which the organs appear to be more drawn 

 out in breadth or in length we cannot assert this without further inquiry. If the 

 organs have an elliptic transverse section the following considerations will lead us 

 to a solution of the problem. We can imagine an elliptic system arising if we 

 project upon an oblique plane the scheme that we construct for organs with 

 circular cross-section. If we consider, for example, the shadows of a circular 

 system which are projected by the sun's rays we can easily satisfy ourselves that 

 upon inclined projection-planes the circles pass over into ellipses of similar excen- 

 tricity. The angle formed by the rafters suffers in this way important changes; 

 in transversely-placed ellipses the height of apex is diminished, in erect ellipses 

 it is increased; the lateral oscillations however remain the same in both cases 

 as they are in circular organs. The same is true for other closed figures of 

 regular form so long as the transverse axes are placed horizontally. The lateral 

 displacements in the case of axes lying obliquely exhibit on the other hand small 

 deviations. Still in this case also the chief character of the oscillations remains 



