APPLICATION OF VORTEX CONSTRDCTION. 57 



(5) The concentration system adopted is the best possible under 

 the circumstances ; and 



(6) As indicated by the proximity of the numbers, 34, 55, the 

 nearest approach to the symmetrical condition, when such a 

 summation series of constants has to be followed. 



In the same manner, making use of the same curve, and dividing 

 the circle into the appropriate number of parts, the other systems 

 for the capitula of Helianthus may be plotted out, within the error 

 of drawing, and numbered up with identical results following the 

 geometrical construction; that there are no members radially 

 superposed to form a true " orthostichy " line, so that each one of 

 the constructions naturally fulfils all the demands of Hofmeister's 

 law. It will be noticed, however, that with a constant direction 

 for the curves, the " genetic spiral " does not run the same way, so 

 that while in fig. 25 (Ps. = 34 + 55) the genetic spiral was right- 

 handed, in similar constructions for (55 + 89) and (21 + 34) it will 

 work out left-handed. Also, the lowest number always gives the 

 number of the longer curves, since in constructions in which the 

 longer curves are formed with the higher numeral the meshes, 

 although orthogonal, are not " squares." 



Taking these Sunfiower heads alone as matured structures, it 

 appears, then, to be evident that the axes bearing them have 

 impressed on them, at an early stage of their development in the 

 first zone of growth, a certain fixed ratio of curves which follow the 

 lines of equal action in the semi-fluid protoplasmic mass, the trans- 

 verse components of which may be represented by the construction 

 of a similar number of orthogonally intersecting logarithmic spirals. 



The numbers of the curves employed and their ratio appear to 

 be an inherent property of the protoplasm of the plant apex, and 

 may vary from shoot to shoot, but within the same capitulum the 

 phenomena remain constant, except in so far as they may be 

 disturbed by secondary changes due to unequal rates of growth of 

 the members composing the system. 



The numbers of the parastichies are the only constants which 

 define the system ; since, although the same points of intersection 

 may be plotted out by other related systems of spirals, it can readily 

 be proved by constructing the diagrams that the curves indicated 



