Manchester Memoirs, Vol. Ixiv. (1920) No. 2 9 



Summary and Conclusion. 



It has been shown that : — 



r. Length-Width and Length-Depth solid distribution figures 

 strongly confirm the view that Retlcularia lineal a (Martin) from 

 braohiopod-beds near Castleton, Derbyshire, is a perfectly bomo- 

 geneou.; species. 



2. Throughout life the length (/) from the brachial umbo to 

 the anterior margin and the width (zu) are connected by the rela- 

 tion: — w=c t {I — c 2 ), where c x and c 2 are constants. 



It is suggested that c 2 is equal to the mean difference 

 between the length from the brachial umbo and that from the 

 hinge-line, to the anterior margin. 



3 Simultaneously the depth (D) and length are related by 

 the expression : — 



D =al J r bl 2J r cl 3 , where a, b, c are constants. 



Finally, the writer would like to express his cordial thanks 

 to Dr. G. Hickling, F.G.S., and to Mr. J. Wilfrid Jackson, F.G.S., 

 for the kind interest they have taken throughout in the work; 

 to Mr. J. Harwood, B.Sc, to whom the author is indebted for 

 the photographs of the solid figures, and for considerable assis- 

 tance in checking many of the calculations incidental to the 

 analysis of the results; and to Professor Sydney Chapman, M.A., 

 D.Sc, F.R.S., for the interest he has shown in the mathematical 

 side of the work. 



LITERATURE. 



1. E. R. Cumings and A. V. Mauck. — " A Quantitative Study of 



Variation in the Fossil Brachiopod Plah 'strophia lynx." American 

 Journal of Science, Ser. 4, Vol. 14, New Haven, 1902 ; p. 9. 



2. Henry Day. — "Variation in a Carboniferous Brachiopod — Retlcu- 



laria lineata (Martin). Manchester Memoirs, Vol. 59, Pt. 1, 

 1915; No. 4. 



3. C. C. Mook. — "Statistical Study of Variation in Spirifer mucronatus" 



Annals , New York Academy of Sciences, Vol. 26, June, 1915 ; 

 P- *1S' 



4. W. E. Alkins. — " Notes on the Growth and Variation of Unto 



pictorum, Linne. Read before the Conchological Society, May 

 14th, 1919 (not yet published). 



Metallurgical Dept., 



The University, Manchester, 

 October 30th, 1919. 



