126 THE MOLECULAR ARCHITECTURE OF PLANT CELL WALLS 



will give arcs which are nearly equatorial. Their actual positions may 

 be determined in the following way. Figure 45 shows the projection 

 circle and pole figure appropriate to the case, the symbols having 

 the meanings already defined. The relation cos y"= tan S tan 6 is 

 obtained from the spherical triangle MPX. For the arcs given by the 

 walls A and B normal to the beam the angular distance of each from the 

 meridian is y)^=S. One has to compare ■yj" with y/-. This can be done 



with specific examples and some of these are shown 

 graphically in Fig. 46, where the larger solid circles 

 correspond to ip^ and the smaller ones to ^". Even 

 with a spiral as steep as S=30° the arcs corre- 

 sponding to the lateral walls are displaced by 7° 

 from the equator and therefore separated by 14°. 

 Sufficient dispersion would cause these arcs to 

 overlap along the equator, but these fused arcs 

 would then of necessity be very diffuse, ranging 

 over an angular distance of the order of 35°. 

 Except, therefore, for the steeper spirals one should 

 not expect a sharp equatorial arc from the lateral 

 walls; and in these steeper spirals an arc of this 

 type would probably be masked by the arcs from 

 waUs A and B which then approach the equator. 



Before proceeding to examine the diagrams of 

 wood in the light of this discussion of the fibre 

 diagram, it will be as well to compare the predic- 

 tions with the actual diagrams of spirals of known 

 spiral angles. A series of such diagrams are pre- 

 ^'^atton.^ee Sl^"' rented in Plate VI. These were prepared for me by 



Dr. V. Ranganathan of the Forest Research Insti- 

 tute, Dehra Dun, India, while working in my laboratory, in the following 

 way. Filaments of artificial silk with remarkably perfect orientation 

 (see Plate VI, Fig. 1) were carefully wound around a horse-hair stretched 

 between the manuals of a micro-manipulator in such a way that the 

 angle between the filament and the hair remained constant over a 

 distance of several milHmetres. In this way a series of spirals with 

 known spiral angles of approximately 0° (parallel bundle), 10°, 20°, 

 30°, etc., up to 90° (flat spiral) were prepared, each of them less than 

 0-5 mm. in diameter and therefore completely covered by the beam 

 issuing from the conventional 0-5 mm. slit in an X-ray spectrometer. 

 The diagrams of these spirals were obtained in turn, with the beam as 

 usual normal to the lengths of the spirals. It will be seen that, in the 



