368 THE REV. H. MOSELEY ON THE GEOMETRICAL FORMS 



To determine the Area of the Surface and the Centre of Gravity of a Turbinated Shell, 

 and the f^olume and Centre of Gravity of its Contained Solid. 



The generating curve of a turbinated shell remains similar to itself as it revolves ; 

 the statical moment of its perimeter varies therefore as the square, and the moment 

 of inertia of its perimeter as the cube, of any of its linear dimensions. In like manner 

 the statical moment of the area of its generating curve varies as the cube, and the 

 moment of inertia as the fourth power, of any of its linear dimensions. 



If therefore C^ €3 C3 C4 C5 Cg represent certain constants determined by the geo- 

 metrical conditions of the generating curve, 



N = CiR2 K = C2R3 H = C3R3 

 M = C4R3 I =C5R4 L = C6R1 

 Therefore the surface of the shell is represented by the integral 



C^J^RdS (10.) 



The co-ordinates of the centre of gravity of the surface are represented by 



Ci/RrfS ' C^/RdS 



and (11.) 



C3/R' d S 

 ^ + C^fRdS' 



And the volume of the contained solid is represented by 



C^fK^de (12.) 



The co-ordinates of the centre of gravity of the contained solid are represented by 



C5/R4 sin c? e C5/R4 cos e <? @ 



Ce/R^rf© I 



^ "^ C4/R^ d& J 



(13.) 



Now it has been shown that in shells R varies according to the law of the loga- 

 rithmic spiral; so that 



p -Scot A 



— xvq e , 



where Rq is the value of R when = 0, and A is the constant angle which the ra- 

 dius vector of the spiral makes with its tangent, whether it be a plane curve or a curve 

 of double curvature ; whence it may readily be proved that 



^=RcosecA (14.) 



Hence, substituting in the preceding formula, and integrating by the known rules, 

 we obtain, for the surface of the shell, the expression 



|CiRo2secA(2^«-*^-l); (15.) 



