MATHEMATICAL NOTES ON LOG. SPIRAL SYSTEMS. 329 



be used as a Standard of Reference for the comparison of the 

 phenomena actually observed on any given shoot. 



It cannot, perhaps, be too strongly insisted that the log. spiral 

 theory is of value solely in so far that it affords such a standard of 

 reference. The same mathematical conception which assumes 

 the possibility of an abstract uniform protoplasmic growth, also 

 takes cognisance of the fact that protoplasmic growth is never 

 uniform, although the approximation may be very close in certain 

 special cases ; and just how close this approximation may be, the 

 mathematical investigation of the log. spiral systems should help 

 to disclose. 



While, again, the mathematical study of these curves may be 

 fairly regarded as outside the province of the botanist, it is clear 

 that the empirical results obtained in previous pages (Part II.) 

 by means of geometrical constructions, more especially in dealing 

 with the convention of hulk-ratio, will have little value unless they 

 can be checked by mathematical methods. 



Note I. — General Equation to the Ovoid Curve in a Log. Spiral 

 Quasi-Square Mesh — the Quasi-Gircle. 



Taking the asymmetrical construction as more primitive and 

 mathematically a more general case of construction, of which 

 whorled symmetry is only a special case, the system can be dis- 

 cussed mathematically in the following terms : — 



In an m : m network of logarithmic spirals, the equation of one 

 set of spirals may be written, 



% log r = n log c + mQ + {2k—l)w, 

 and of the other, 



m log r = m log c — nQ-\-{2l — l)Tr; 

 where k, I, are any integers, positive or negative. 



Values of k differing by m refer to consecutive turns of the same 

 spiral, and similarly values of I differing by n. 



The " centres of construction " of the network are given by the 

 intermediate spirals, 



n\ogr = n log c -)- m0 + 2fcV, 

 and «i log r = m log c — «0 + 21' ir. 



