126 BULLETIN 82, UNITED STATES NATIONAL MUSEUM. 



In a species having five division series, all interpolated, there are five repe- 

 titions of the first brachial pair, plus the first brachial pair itself, making six 

 homologous pairs of ossicles between the radials and the free arms from the 

 third brachial onward. These may be numbered I, II, III, IV, V, and VI, I 

 representing the IBr series and VI the first brachial pair. 



Ontogenetically these appear in the order I, TV, II, III. IV, V, VP, VI^ 

 being the first brachial pair at the base of the adult free undivided arm, wliich 

 is an exact reduplication of VI at the base of the discarded larval arm. 



If tlie simplest type of arm structure were also the most primitive the phjdo- 

 genetical sequence of the pairs would be 



VI. I, II, III, IV, V, VP, 



since VI is invariably present, whether the others are present or not. But from 

 the all but invariable presence of I in addition to VI in the crinoid arm it is 

 probable that the actual phylogenetical sequence is the same as the ontogenetical 

 sequence, both being 



I, TV, II, III, IV, V, TV^ 



Morpliologically VI is always jjresent. With the appearance of a gap between 

 the radial and the arm base as a result of the shrinking of the calyx the appear- 

 ance of V is necessitated, and subsequently also of IV, III, II, and I, so that 

 the morphological sequence is 



TV, V, IV, III, II, I, 



TV-, being of the same morphological value as TV, not appearing. 



The number of arms on each raj' which would result from the presence of 

 the full complement of division series — a rare condition beyond the IIBr series 

 and quite exceptional beyond the IIIBr series — is, (no division series present), 

 1 ; IBr series, 2 ; IIBr series, 4 : IIIBr series, 8 ; I VBr series, 16 ; VBr series, 32 ; 

 VBr series, 64; VIIBr series, 128; 5-rayed comatulids with these series as just 

 given would have the following numbers of arms: (no division series present), 5; 

 IBr series present, 10; IIBr series, 20; IIIBr series, 40; I VBr .series, SO; VBr 

 series, 160 ; VIBr series, 320 ; and VIIBr series, 640. 



HliACHIAI.S OF THE FREE UXDIVIDBD AHM IIBTOND THE FIR.ST PAIR. 



In the free imdivided arms of the comatulids the first two brachials, which 

 together, as previously explained, form a single unit, are immediately followed, 

 excepting only in the PentametrocrinidiK (fig 126, p. 79) and in Atopocnnus 

 (part 1, fig. 227, p. 245), by a second pair of much shorter brachials invariably 

 united by syzygj-, just as the first tM^o are united by synaitliry (fig. 128, p. 79) ; 

 then comes, except in the Atelecrinidse, a series of short wedge-shaped brachials 

 which are commonly about twice as wide as the median length; after usually 

 between the ninth to the twelfth the brachials typically become triangular and 

 about as long as broad (fig. 233, p. 193). after the end of the proximal third or 



