Curves formed by Cephalopoda and other Mollusks. 253 



more convenient to compare the outer edge with the intersec- 

 tion of the whorl with the one next to it inside. 



To determine the angle of retardation of the inner edge behind 

 the outer in a discoid shell. 



This element is given by the ratio of the umbilicus and of 

 the outer whorl to the diameter. If \ and \i be these ratios, 



1 p— jScota 



whence 



/31oo*R = — 7rloo'X and R = ■= — > 



& 8 1 — X — fju, 



or _ it log X 



H = 5 



log(l— X— /*) — log ft 



In the description of Cephalopoda we often meet with such 

 an expression as "inner whorls two thirds concealed." If the 

 inner whorl be concealed in the ratio of x : 1. 



1-R 2 \ 



x = 



■27rcota e (-|3-7r)cota J J{^ > 



whence, if R is known, X and then ft may be deduced ; which 

 might sometimes be convenient in the case of a fragment. In 

 this case, however, another method may be followed. If b ± 

 and b 2 be the lengths of two lines measured in the direction of 

 the radii, and s the arc of the outer curve between them, 

 then 



b ± — b 2 = s cos *(l-e-P cotCi ). 



As an example Ammonites margaritatus may be taken. In 

 one diameter of the septate portion of the shell, of length 166^ 

 millims., the umbilicus was 51 and the last whorl 72 ; this 

 gives X = *307 and fi = '432. In another perpendicular dia- 

 meter of 131 millims. the umbilicus was 42 J and the last 

 whorl 55 ; whence X = *324 and //, = *41. The values of R de- 

 duced from these are 1*65 and 1*64. The decrease in the value 

 of X and the corresponding increase of /jl shows that this spe- 

 cies grows more involute with age ; in other words, B varies. 

 In the earlier whorls it is 820°. It might be thought that 

 Professor Naumann's conchospiral would in this instance pro- 

 duce closer approximations to the observed values ; but the 

 value of R deduced from the radii at right angles is 1 # 69 ; 

 whereas it ought to be less than that derived from whorl- 

 breadths, if the inner and outer curve were both conchospirals. 

 Another example taken was Turrilites scheutzerianus. Here 

 the diameter at the last whorl was 56 millims., the penultimate 

 diameter 43 millims. ; the last whorl had a horizontal breadth 



