1046 THE THEORY OF TRANSFORMATIONS [ch. 

 9:1, and the inner curve as 3 : 1 ; while the five petals become separate 



when a = b, and the formula reduces to r = cos^ — . 



z 



In Fig. 504 we have what looks like a first approximation to a 



horse-chestnut leaf. It consists of so many separate leaflets, akin 



to the five petals in Fig. 503; but these are now inscribed in (or 



have a locus m) the cordate or reniform outline of Fig. 500. The 



new curve is, in short, a composite one; and its general formula is 



r = sin 6 12. sin nd. The small size of the two leaflets adjacent to 



the petiole is characteristic of the curve, and helps to explain the 



development of "stipules." 



Fig. 502. Grandi's curves based on 

 r = sin5^, and illustrating the five 

 petals of a simple flower. 



Fig. 503. Diagram illustrating a corolla 

 of five petals, or of five lobes, are 

 based on the equation r = a + b cos 6. 



In this last case we have combined one curve with another, and 

 the doing so opens out a new range of possibilities. On the outhne 

 of the simple leaf, whether ovate, lanceolate or cordate, we may 

 superpose secondary sine-curves of lesser period and varying ampli- 

 tude, after the fashion of a Fourier series ; and the results will vary 

 from a mere crenate outline to the digitate lobes of an ivy-leaf, or 

 to separate leaflets such as we have just studied in the horse-chestnut. 

 Or again, we may inscribe the separate petals of Fig. 505 within a 

 spiral curve, equable or equiangular as the case may be ; and then, 

 continuing the series on and on, we shall obtain a figure resembling 

 the clustered leaves of a stonecrop, or the petals of a water-lily or 

 other polypetalous flower. 



