202 On the Measurement of the Curves formed hy MoUusks. 



whence 



tan co = cos a tan i (1) 



If x be the angle between the tangent and the radius from the 

 vertex. 



dz sec co = da cos % : 



.*. cos co = cos *'sec^ (2) 



The values, therefore, of e and ^ are sufficient to give co 

 and a. This last equation was obtained by Professor Moseley 

 bv a different method, and the first, under a different form, de- 

 duced from it. It follows from (1) that i increases with co in 

 the same shell, and hence that the angles must be measured at 

 corresponding points. In the same way % is dependent on co, 

 since from (1) and (2) we may deduce 



tan x = tan a sin co (3) 



The angle % is the only one which can be well measured on 

 specimens, and i only on figures. The " sutural angle'' em- 

 ployed by D'Orbigny is different from either of these, being 

 the angle between the line joining the ends of two radii cor- 

 responding to 6 and 6 + tt with the smaller of them. The 

 connexion of this with the drawing of the suture is only ap- 

 proximate, as the latter does not lie in one plane. If ^ be 

 this angle, we have 



,,cot a= ^ ^ 

 sin (\jr + 2co) 

 or 



. R sin 2co 



tan^ = - ^5 — U) 



T 1 — Kcos2w v ' 



If &) be found from the tangent lines to the shell, yfr must be 

 measured from the points of contact and not in the suture — 

 whence the term " sutural " angle is inexact for a second 

 reason. This method has the advantage, however, of being 

 applicable to very little more than a complete whorl, the rest 

 requiring one and a half. "When less than one whorl is pre- 

 served, it is sometimes possible to measure the angle between 

 the projections of two tangents to the curve at points separated 

 by half a whorl, on a plane through the axis perpendicular to 

 that containing them, i. e. between BG- (fig. 10) and the corre- 

 sponding line on the opposite side. If 2cp be this angle, we 

 have 



x dz 

 tan ©= — 77,= cot co tan a. 



rau 



One important value of these equations is to serve as a 

 check on the separate measurements, and thus to gain obser- 



