454 Proceedings of the Poyal Society 
handed — spiral. 
Zjl 
At the base of the cone, however, a number of 
rudimentary scales of small size and somewhat peculiar shape are 
intercalated with considerable regularity among the others, so as to 
appear as projections placed at the intersections of the lines formed 
by the margins of the larger scales. Now, if these small scales had 
been disposed with perfect regularity, and had been of equal size 
with the others, there would have been a left-handed bi jugate 
arrangement, with divergence 
Such a cone, in fact, sug- 
21x2 
gests the possibility of single spirals of the ordinary series being 
derived from bijugates of the same series by suppression of one 
half of the scales. 
Again, the ordinary trijugates are easily derivable from bijugates, 
as indicated in Table E. 
Table E. — Showing the possible derivation of ordinary Trijugate from 
the Bijugate Arrangement. 
DSD 
— 3 6 
2 4 6 
From the ordinary trijugate, in turn, a spiral of the system, -, 
2 3 5 
- , — , — , &c., may be simply derived, as indicated in Table E. 
y 1 4 Zo 
s 
9 
10 
V 
15 = 
16 = 
2 
5x3 
3 
8x2 
Table E. — Showing possible derivation of a Spiral of the System, 
1 1 
4’ 5’ 
&c., from the Ordinary Trijugate. 
D 
S 
D 
S 
D 
S 
V 
1 
4 
5 
9 
14 
23 
37 = 
_ 
3 
6 
9 
15 
24 
39 = 
Again, it is clear that by augmentation of parts, a spiral of the 
112 
system - , - , - , &c., may be derived from the ordinary bijugate, 
o 4 7 
since the converse (by diminution) actually occurs in the second 
of Mr Smyth’s cones indicated in Table B. 
