124 THE MOLECULAR ARCHITECTURE OF PLANT CELL WALLS 



since at these points the intensity is theoretically infinite. This means 

 that the intensity is the greatest at the ends of the arcs. Each arc 

 therefore breaks up into two spots and the diagram consists of four 

 spots, symmetrically placed, instead of two lateral arcs. If to this be 

 added the angular dispersion of the micelles it is seen that the spiral 

 photograph is liable to be somewhat complex. Depending on the value 

 of the S, the four arcs may overlap on the equator, giving a false 

 impression of a real equatorial spot, or even, if the dispersion is great 

 enough, fuse into two meridional arcs. Finally, overlap may occur in 

 both positions, giving eight apparent arcs in all, only four of which are 

 real. Several further points appear from the equation for intensity: 



(1) Provided the dispersion is not unduly high, a fibre of circular 

 cross-section gives a photograph in which the expected four arcs appear 

 symmetrically placed. 



(2) As S, the spiral angle, increases, the four spots approach the 

 meridian more and more closely. As this happens, the danger of an 

 overlapping of these arcs along the meridian, giving a spurious arc, 

 becomes greater. At a value of S given by cos 5= sin 6, the spots fuse 

 into two meridional arcs (because at this value of 5", 7^= oo only when 

 ^=0 or 180°). Taking the planes of 3-9 A., 5-4 A. and 6-1 A. spacing, 

 the critical values for S appear to be as follows: 



S (limiting) 



Spacing No dispersion Dispersion ±5° 

 3-9 78-5° 73-5° 



5-4 81-8° 76-5° 



6-1 82-8° 77-5° 



Hence all fibres whose spirals are flatter than 5= 73-5, say, for the 3-9 

 arc, or than the corresponding figure for the other two arcs, will give 

 two meridional spots instead of four lateral arcs. In a mixture of fibres 

 in which 5 varies sufficiently widely, therefore, one would expect spirals 

 flatter than those given in the above table to contribute to the meri- 

 dional arc, and the 3-9 arc should therefore appear intense out of pro- 

 portion to the 5-4 and 6-1 arcs. It is interesting to note that this is the 

 case here, and that in tracheids generally spirals do occur with sufficiently 

 great values of S. 



(3) If the spiral is flat, i.e. 5=90°, then the intensity equation reduces 



to 



/pOC 1/^(1— cos- 6 cos^ ip). 



At ^=0, therefore, the intensity is proportional to 1/sin 6 and at ^^=90° 



