THE FOSSIL CEINOID GENUS DOLATOCRINUS AND ITS ALLIES. 21 



by this means they were reheved of the trouble of comparison with 

 species otherwise similar, which they usually declared to be unneces- 

 sary. Thus in Bulletin 8, page 46, in describing D. charlestownensis: 

 It is, of course, unnecessary to compare it with any other described species, because 

 the arm formula alone distinguishes it. 



This plan placed the describing of species upon a sure mathematical 

 basis; but it had its limitations. The range of variation in number 

 of arms in a single otherwise constant form of Dolatocrinus mnj be 

 from 10 to 20, which w^as soon in danger of being used up. So the 

 authors w^ent a step farther, and evolved another scheme of infinite 

 possibilities^ namely, that not only is the number of arms an invari- 

 able specific character, but also the mode of grouping of the arms 

 among the five rays — so that newly acquired specimens not other- 

 wise distinguishable from species already described could readily be 

 made the types of new species upon this character alone. Thus they 

 say in the ninth Bulletin, page 47: 



This species is distinguished from all other 16-armed species * * * by the arm 

 formula. For example, the arm formula in D. coelatus is A-A-A-2-2; in D. nodosus 

 it is 4-3-3-3-3; in D. salebrosus it is 4-3-2-3-4; in D. arroms it is 4-4-3-3-2. 



x\nd on page 51 : 



It can be of no service to compare this with any species having a different arm 

 formula, for that alone distinguishes it. 



The beauty of this plan will be evident when we consider that if 

 we take only the most usual variations, caused by the addition of 

 one or two arms, thus giving 2,3, or 4 arms to the ray, there would 

 be 120 different permutations of these numbers among the five rays 

 of a crinoid, each of which under the authors' latest rule would estab- 

 lish a distinct species; that is to say, 120 species under any otherwise 

 defined form. Every one of the dozen or more recognizable Hamilton 

 forms of that locality may exhibit similar variations in number and 

 arrangement of arms; so that with sufficient collections there would 

 be as the logical result of this process, consistently followed up, about 

 1,500 possible species of this one genus, in a single formation S feet 

 thick, and at the same locality. 



Now it is a fact abundantly established that a character which 

 is available for the differentiation of species in one group or genus, 

 or in one formation or locality, may as a result of different conditions 

 be utterly worthless in others. This is notably true in regard to the 

 number of arms in the crinoids. Among the recent forms, especially 

 the comatulids, variation from 10 to 20 arms is not uncommon, while 

 on the other hand a definite number of arms is constant for many 

 large groups. 



In certain prolific and extremely well-marked Carboniferous spe- 

 cies, such as Macrocrinus verneuilianus and Dizygocrinus rotundus, 

 there is a considerable range of variation in number of arms (which 



