Curves formed by Cephalopoda and other Mollusks. 243 



part of the description, when complete, of every regular mol- 

 luscan shell. 



3. As an example, suppose the trace elliptical, which is very 

 often the case. 



To find the equation to the surface of a turbinated shell of 

 elliptical section. 



Taking the pole as origin, and the axis of the shell as that 

 of z, let X, /j, be the radial and vertical coordinates of the centre 

 of any one of the elliptic sections ; a, b the semiaxes ; e the 

 angle the major axis makes with that of z. Then the equation 

 to the ellipse is 



Uz—fjb) cos e+ (r— X) sin ej- 2 -j(V — X) cos e — (z— yu)sin ej- 2 



a 2 + P ' 



Taking the centre as the point of reference, if y be the angle 

 of elevation, 



/ tt = Xtany; 



and if a be the spiral-angle, 



X=e 0cota . 



Also, if ft be the angle of retardation of the inner end of the 

 major axis behind the centre, 



a= ^cota__g<0-0)cotaj. cosec e= e 0cot a (l — e~^ cota ) cosec 6 



= e 6 cot a cr cosec e, say. 



Putting b = /ea and making the above substitutions, the equa- 

 tion to the surface becomes 



fc 2 \(z— e 6cotac tan 7) cos e + (« — e e cota ) sin e} 2 



+ \(z — e* cot a tan y) sine — (r— e* cota )cosep 



= /e 2 <7 2 cosec 2 e.e 2flcota , (A) 



which may be immediately transformed into one between x,y,z 



_j y 

 by the substitution of x 2 -\-y 2 for r 2 , and tan * for 0, or into 

 a polar one by substituting p cos (/>, p sin cf> for z and r ; which 

 will bring it into harmony with the more general functional 

 but unmanageable equation given by Professor Moseley. In 

 most cases, except among the Gasteropoda, e may be assumed 

 = 90°. 



4. By giving different values to the constants in this equa- 

 tion we can obtain all the varieties of shells of elliptical sec- 

 tion, and in a similar manner those of any other form of 

 section. Thus for all the Brachiopoda the value of y is zero. 

 In both these and the Lamellibranchiata a is never much less 

 than unitv, and often greater ; and at the same time a is small 



R2 



