370 THE REV. H. MOSELEY ON TURBINATED AND DISCOID SHELLS. 



^^ (g(e-2„,)cotA_ 1) ^ ^^^(e- 2.) cot A. ^L______i^ ^. !_ (^^) (,4ecotA_ 1 ) (22.) 



7b determine the Polar Equation to the surface of a Turbinated Shell. 



Let m (fig. 4.) be any point in the surface of the shell, and let the equation to the 

 curve V Q T, imagined to be in the act of generating the point m, be 



where B is an arbitrary constant representing a linear dimension of the curve, and 

 therefore varying according to the law of the logarithmic spiral, so that it may be 

 represented by the formula 



where Bq is the initial value of B. Suppose the abscissae of the curve to be measured 

 along the axis V T from V, so that V w and u m are co-ordinates of m. Let A w = f , 

 mA^ = 0, Aw = f cos <I> = Zi + a?i, M m = f sin O = ^1 



.-.^sinO) =/(Bog«'=°*^f cosa>- 2i) (23.) 



or substituting for ^i its value 



fsm<l'=/{B„e«"'*,?cos<t - r„(et»-^»'' "'*-!) - r,8(^-°'")«"^( '^_,^,„.^-; ) j (24.) 



From the above may readily be determined the equation to the surface of a shell 



between the rectangular co-ordinates x, y, z. Observing that — 2 w ^ = tan" ^ —, 



and substituting, we obtain 



^It^ (2n«r + tan-»|-)cotA (tan-»-^cotA ) tan-»^cotA / - 2n«-cotA__ ,\ 1 M25.) 



The values of the constants C^ Cg C3 C4 C5 Cg are dependent upon the geometrical 

 form of the generating curve in each particular shell ; the constants Rq r^ and B^ on 

 its dimensions at the point where the generation of the shell is supposed to com- 

 mence. 



The constant A is independent of the form and dimensions of the generating curve. 

 It depends simply upon the law of that particular logarithmic spiral which is affected 

 by that species of shell. 



To determine the Constant Angle of the Spiral affected by any given Shell. 



The common ratio of the geometrical progression according to which the widths 

 of successive whorls increase being determined by actual admeasurement and repre- 

 sented by X, we have the equation 



g2 it cot A J, 



•••A = tan-(i^J (26.) 



