101 



depressed in various portions of their course ; it will be with 

 these flexuosities, generally inconsiderable, as with those 

 treated of above in the case of the cylinder, and we shall still 

 be right in concluding generally on the constancy of the primi- 

 tive divergence from that of the secondary divergence. And 

 finally, in the general case, when the place of the insertions is 

 conical, nothing prevents our conceiving it to be divided into 

 horizontal sections, each of which would belong to various 

 cones ; and our results, true for each section, must also hold 

 good for the whole. 



With respect to the parallelism and equidistance of the se- 

 condary spires among themselves, this is a supposition ren- 

 dered sufficiently evident by the direct observation of the plant; 

 it is only occasionally changed by that force which at times 

 disturbs the normal height of the insertions ; and as soon as 

 this normal height is re-established, the parallelism again 

 makes its appearance. 



These results (those namely of the preceding paragraphs) 

 may be considered as general and independent of the geome- 

 trical form of the place of the insertions, and even of the re- 

 gularity of the secondary spires. 



The torsion of the stem must however not be neglected : 

 sometimes it takes place in an opposite direction to the pri- 

 mitive spire, and then diminishes its divergence in appearance ; 

 at times it occurs in the same direction, and in this case the 

 primitive divergence is on the contrary increased. This force 

 rarely produces, in the opinion of the authors, any great effects; 

 nevertheless it is highly important, in all cases where it is pos- 

 sible, to take into account this source of error. 



The second chapter of this memoir contains the special con- 

 sideration of all the cases occurring in the curviseriate leaves, 

 to which I must refer the reader. The general results are how- 

 ever the following. 



1. When an aggregation presents several spirals whose se- 

 condary numbers are prime to each other, the insertions are 

 arranged on a single primitive spire, and are separated from 

 each other by a constant divergence from one extremity of 

 the aggregation to the other. 



2. If the secondary numbers have 2, 3, or 4 for their com- 

 mon divisor, the insertions are arranged in whorls of 2, 3, 



