330 RELATION OF PHYLLOTAXIS TO MECHANICAL LAWS. 



To find the curve inscribed in a mesh of the network analogous 

 to a circle inscribed in a square, use the orthogonal transformation 

 log r = x, Q = y : also put log c = a. 



The equations of the two sets become, 



nx — my = na-\-{21c — l)ir, 

 mx+ny = 'ma + {2l—l)Tr, 



so that in the transformed system the meshes are equal squares of 



side ^^ 



Consider the mesh whose sides are given by A = 0, ^ = 1 ; 



Z = 0, ^ = 1. 

 Its " centre of construction " is at the intersection of the spirals 

 given by k' = 0, r = 0, i.e. is at the point r = c, 6 = 0. 



The sides of the corresponding square in the transformed system 

 are, 



nx — m/y= wa±7r, 



and its centre is at the point x = a, ?/= 0. 



The equation of the circle touching the sides is 



Transforming back, the equation of the required curve is found 

 to be : — 



\2 



('^9' 





The logarithm is the natural logarithm, and Q is measured in 

 circular measure : when the logarithm is the tabular log and Q is 

 measured in degrees the equation may be written : — 



log r = log c±l-36438/.^^- -00003086402. (H.) 



The point corresponding to the centre of the circle is ?' = c, = 0, 

 i.e. is the " centre of construction.'' 



Since all the meshes are similar, differing only in size, the above 

 equation will apply to any mesh, if c is the distance of the centre 



