OF TURBINATED AND DISCOID SHELLS. 369 



(16.) 

 (17.) 



for the CO ordinates of the centre of gravity, 



2 Cg Rq (3 cot A sin @ - cos O) e^ecot A ^ ^ 



Ci(tanA + 9cotA)* g2ecotA_ ^ 



QCgRp (ScotAcose + sin@)63Q<^«tA_3cotA 



Ci(tanA + 9cotA) * g2ecotA_ j *. * 



/e— 2n?rcotA i\ o /fj P \ /.3ecotA_ i\ 

 /,(e-2n*)cotA_ ,x , e-2*)cotA. (f Zil] A. — l^:^^^) (- Zl] /'^«^ 



Observing that r^ being taken to represent the initial length of the lesser diameter* 

 VT, of the generating curve, Tog® *^°*'^ will represent the length of that diameter after 

 the generating curve has revolved through the angle 0, and ^q eC®-^'^''"*^ will repre- 

 sent the width of the next preceding whorl of the shell, measured in the direction of 

 this diameter produced ; and the sum of the widths of all the preceding whorls, sup- 

 posed to be n in number, and measured in this direction, will be represented by 



■Kn „ .(e — 2 n a-) cot A 



Moreover, that the lesser diameter sliding along the axis, as the curve revolves 

 through any angle, a distance precisely equal to that by which the diameter increases, 

 it follows that the distance from the edge of the last or wth of the preceding whorls, 

 measured in this direction, to the origin is represented by 



^jg(e-2n.)cotA_ 1)^ 



So that «^-}~ is represented by the formula 



^ /g(e-2n!r)cotA j\ _j_ 2" r g(® - 2 w *) cot A^ 



Integrating the formula (12.), having substituted for the value of R, we find for the 

 VOLUME of the solid contained by the shell the expression 



yC4Ro3tanA(g^«^°*^- l).t (19.) 



And integrating the formula (13.), the co-ordinates of the centre of gravity of the 

 contained solid are found to be 



3C5R0 (4cotAsin@-cos@)64^^°tA ^ ^ 



C4(tanA + l6cotA) ' g3ecotA_ j • • • • V -J 



3 C5 Rq (4 cot A cos © + sin ©) e'* ® ~* ^- 4 cot A 



C4(tanA+ leicotA) g3ecotA_ ^ 



(21.) 



* "When the whorls partially overlap one another, this diameter is to be understood to extend only across 

 that portion of the generating curve which actually generates the chamber of the shell, and which is not inter- 

 fered with by the preceding whorl. In these cases, then, it will only be a portion of what would be the shorter 

 diameter of the generating cur^^e, if that curve were completed. 



t In the case in which the generating curve does not slide upon the axis as it revolves, Zi = 0. 



X In the case of turbinated shells R^ may be considered extremely small with respect to any existing di- 

 mensions, and exceedingly great, so that the formula 19. being taken to represent the whole capacity of the 



shell, becomes in this case — C4 Ro^ gS ecot A^ ^^^^ varies as R'. 

 o 



MDCCCXXXVIII. 3 B 



