36 Prof. Tait on Listing's Topologie. 



not only because of the different temperatures of its ends, but 

 because of the differences of their electric potential (due to the 

 " Thomson effect"). 



The same is generally true of every vector (or directed) 

 quantity, such as a velocity, a force, a flux, an axis of rota- 

 tion, &c. 



(6) An excellent example of our science is furnished by 

 the Quincunx, which is the basis of the subject of Phyllotaxis 

 in botany, as well as of the arrangement of scales on a fish. 



A quincunx (from the scientific point of view) is merely 

 the system of points of intersection of two series of equidistant 

 parallel lines in the same plane. By a simple shear parallel 

 to one of the two series of lines, combined (if necessary) 

 with mere uniform extensions or contractions along either or 

 both series, any one quincunx can be changed into any other. 

 Hence the problems connected with the elements of the sub- 

 ject are very simple; for it follows from the above statements 

 that any quincunx can be reduced to square order. The 

 botanist who studies the arrangement of buds or leaf-stalks 

 on a stem, or of the scales on a fir-cone, seeks the fundamental 

 spiral, as he calls it, that on which all the buds or scales lie. 

 And he then fully characterizes each particular arrangement 

 by specifying whether this spiral is a right- or left-handed 

 screw, and what is its divergence. The divergence is the angle 

 (taken as never greater than it) of rotation about the axis of 

 the fundamental spiral from one bud or scale to the next. 



(7) It is clear that if the stem or cone (supposed cylin- 

 drical) were inked and rolled on a sheet of paper, a quincunx 

 (Plate II. fig. 1) would be traced, consisting of continuously 

 repeated (but, of course, perverted?) impressions of the whole 

 surface. Hence if A, A 1 be successive prints of the same scale, 

 B a scale which can be reached from A by a right-handed spiral, 

 A B, of m steps, or by a left-handed spiral, A 1 B, of n steps, 

 these two spirals being so chosen that all the scales lie on n 

 spirals parallel to A B and also on m spirals parallel to A 1 B, 

 we shall find a scale of the fundamental spiral by seeking the 

 scale nearest to A A x within the space A B A x . 



Here continued fractions perforce come in. Let/*,/*/ be 

 the last convergent to m / n. Then, if it be greater than m / n, 

 count fju leaves or scales from A along A B, and thence v 

 leaves or scales parallel to B A 1; and we arrive at the required 

 leaf or scale.- If the last convergent be less than m/n } count 

 v leaves along A 1 B, and thence //, parallel to B A. If the 

 leaf, a, so found in either case, be nearer to A than to A v the 

 fundamental spiral (as printed, i. e. perverted) is right-handed; 

 and vice versa. Thus the first criterion is settled. 



