OF TURBINATED AND DISCOID SHELLS. 363 



breadths, of which those of the surface described by P C will be the least, and those 

 described by Q C the greatest. 



.•.^AS<AU<g±^.AS. . 



Considering, therefore, S, U, and N as functions of R, expanding by Taylor's theorem 

 and dividing by A R 



n' dR"^ dR^'R' l.Q'^^^-'^dR'^ dR^' 1.2'^^^'^R'dR 



+ 



CdS fl f/N l\ . 1 ^'S1 . „ , . 



The second of these series having, for all values of A R, a value intermediate between 

 the other two, and the first terms of these other two being equal ; the first terms of 

 the three series are equal. 



^ ^_ N ^ 



•'•<^R"~R*rfR 

 and 



m-'iR (2-) 



^=fl 



which to adapt it for integration, (R being a function of 0) may be put under the 



form 



,, /-N dS dR ,_ 



The expression for the area of the surface thus determined, on the supposition that 

 the generating curve does not alter its position in respect to the axis otherwise than by 

 revolving round it, is the same with that of the surface which would be generated by a 

 curve which, as it revolved about the axis, slided along it, a different form being as- 

 signed to the function N. For if we imagine a conchoidal surface of this general 

 form (fig. 4.) to be intersected by planes, exceedingly near to one another, passing 

 through its axis, and at the same time to be traversed, as the surfaces of turbinated 

 shells usually are, by spiral lines parallel to the direction of the whorl, and which may 

 be understood to mark the paths of given points in the generating curve*; then each 

 element of the surface intercepted between two of the planes spoken of will, by these 

 spiral lines, be divided into a series of oblique parallelograms, two adjacent sides (con- 

 taining an acute angle) of each of which, may be considered as intersections with the 

 conchoidal surface of two planes, which intersect one another in an ordinate of the 

 generating curve ; one of these planes is a tangent to one of the spiral lines spoken 

 of, and the other is the plane of the generating curve itself. Now let us suppose the 

 inclination of these planes to one another to be constant, as is always the case in 

 shells, and let it be represented by A. Let moreover the inclination, to its ordinate, 



* This demonstration will be best understood by referring to the actual surface of a turbinated shell on 

 which the spiral lines are visible. 



3 a2 



