258 DE. G. H. FOWLER— BISCAYAN PLANKTON: 



percentage of the measurements involved *. The present work, based on such unfor- 

 tunately small forms as Ostracoda, can at best be merely pioneer work, and cannot 

 pretend to mathematical accuracy. The means here given can only be approximations 

 in most cases, owing not only to the minuteness but also to the scarcity of specimens. 

 Although in tliis case unavoidable, it is, mathematically speaking, an absurdity to treat less 

 than a hundred cases for problems of frequency. If the means are only approximate, 

 so also must be the growth-factors. Eurther, it is possible that not only are the means 

 and growth-factors mere approximations, but also the law itself, as here phrased. The 

 actual increase at each moult is probably not so simple that the same growth-factor will 

 apply exactly at every moult, even in Ostracoda of small size and few stages ; and, as 

 will be seen later in a discussion of measurements of crabs and lobsters, the factor 

 certainly undergoes modification as growth proceeds, in largo forms with numerous 

 moults. On the other hand, I have little doubt that the law as phrased on p. 224 is a 

 sufficiently accurate weapon for the zoologist to use in combination with the morpho- 

 logical evidence, even if the words ^'^ fixed percentage " may require revision at the 

 hands of the mathematician. 



After writing the larger part of tliis paper, I had the great advantage of the iielp 

 and criticism of Professor Karl Pearson from the mathematical standpoint. He 

 expressed the opinion that while valid statistical evidence is given here in many cases 

 for that separation of the stages which I have proposed, in others this evidence is 

 wanting [in some of these latter, however, it is supplied by tlie morphological changes 

 which indicate that an intermediate moult has occurred]. Professor Pearson, regarding 

 the " staircase-growth " of Crustacea [discontinuous growth by moults] as likely to fit in 

 eventually with the laws of growth of other organisms, pointed out that Brooks's law 

 could only fit in with a relatively small portion of the general growth-curve ; that it 

 might be a reasonable approximation to a portion of it, but that mathematically it is 

 clear that many other formuliB would accomplish the same result ; that in most cases 

 I have only recorded three stages, and liave accordingly only two ratios of growth, and 

 that if these were not very widely different one might easily get a mean value within 

 the j)robable error of the results. In sum, he was not convinced of my conclusions 

 being more than a first approximation based on three stages. With this verdict I am 

 content. I believe that I have carried the matter about as far as Ostracoda will admit, 

 and must leave to others the further development of the suggested law of growth on 

 more favourable material. 



It is reasonable to suppose that all the species of Coiicliwcia follow the same rhytlim 

 of moults ; and on this is based the working supposition that they all exhil)it two sta,ges 

 with secondary sexual characters in the male. These two stages have been recorded 

 (in one or both sexes, in this or other memoirs) i'or citrta, daphnokles, elegans, haddoni, 

 hycdojjhyllnm, loricata, magna, rotundata, and spiiiirostris (supposing, that is, that my 

 identifications are correct); and their occurrencn^ has consequently been inferred in 

 ametra, ivihricata, procera, rhynchena, and s])inifera. 



* ObsL'iTatious on lobsters iiiid criiljs will In' loiiiiil on \\\\. L'79-l*81. 



