Br. 0. Rernnann — Distribution of GraptoUtes. 453 



"the angle of divergence is gradually carried through the various 

 species from 180° almost to 360^," which is as much as to say that 

 the angle of divergence is the angle formed by the dorsal " non- 

 polypiferous " margins of the bi'anches. 



Several authors, in order to arrive at a definition of the angle of 

 divergence, have started from the genus Dlplograptus, M'Coy (Fig. 5), 

 assumed the angle in question here, as shown in the figure, to be 0°, 

 and then proceeded to Dicranograptns, Hall, Dicellograptus, and 

 Didi/mograptus, and found for these genera the magnitudes 0°, 

 ^180°, z 180°, and 7 180° to Z 360°. 



Another mode of definition seems to me to recommend itself as 

 the more natural. The oldest two-branched Graptolite genus, which 

 first of all induces us to think over the angle of divergence, is the . 

 genus Didymograptus, M'Coy. If we have an example of this genus 

 before us (Fig. 2), we shall find it natural at the first glance to give 

 as the angle of divergence the angle which the branches form on their 

 cell-bearing side.^ 



The younger two-branched Graptolites following Didymograptus, 

 M'Coy, in the sedimentary deposits, the Bicellograpti (Fig. 3) may 

 then be regarded as Didymograpti the branches of which are bent 

 further back. The above-indicated angle between the cell-bearing 

 margins of the branches has become 7 180°. In Dicranograptns, 

 Hall, only the distal parts of the branches are in divergence, and 

 the angle here formed by the cell-bearing margins of the branches 

 (Fig. 4) varies between the same limits as in Dicellograptus, Hopk. 

 The basal parts of the branches in Dicranograptus have grown 

 together by their dorsal margins, so that the cell-bearing margins 

 form an angle of 360°. Lastly, the genus Diplograptus (Fig. 5), 

 which represents the Graptolites in their most perfect development, 

 no longer possesses any visibly diverging branches ; but the theo- 

 retical angle of divergence at which we have arrived step by step, 

 from the genus Didymograptus, through Dicellograptus and Dicrano- 

 graptus, amounts to 360°. In Monograptus, Gein. (Fig. 6), we 

 cannot speak even of a theoretical angle of divergence. 



This conception, which takes into account the geological age of the 

 genera of Graptolites, may perhaps deserve consideration. Under 

 the term angle of divergence we therefore understand the angle formed 

 by the cell-bearing margins of the branches. This is as follows for the 

 genera: — Didymograptus, M'Coj =z 70° to 180°. 



Dicellograptus, Hopk. = 180° to Z. 360°. 

 Dicranograptus, Hall = 360° and 180° to Z. 360°. 

 Diplograptus, M'Coy = 360°. 



^ The question of the angle of divergence appears to me to be a comparatively 

 simple matter. Leaving out of consideration the two sides of the branches, the angle 

 of divergence is that formed by the axial lines of the branches at their meeting-point 

 in the sicula. Assuming the two branches to grow quite straight out from the sicula, 

 there will be no angle of divergence (0°),and the celluliferous margins will bein 

 contact ; — then, as the branches diverge, sweeping round the imaginary circle of which 

 the sicula is the centre (of course in the direction of the non-celluliferous margin, as 

 otherwise they would have to cross each other) , these axial lines will form gradually 

 increasing angles, until the dorsal margins come close together, when we get 3G0° as 

 in fig. 5. Roughly speaking, the angles shown in the figures would be, for fig. 2, 

 about 90° ; fig. 3, about 270° ; fig. 4, 360°, and 330°; and for fig. 5, 360°.— W.S.D. 



