The Rev. H. Moseley on Conchyliometry. 305 



the spiral P Q is described be developed, this spiral will be- 

 come a plane logarithmic spiral, whose polar equation has 

 been shown to be 



R = R g e cot A where — sin i 



Now sin i cot A = — -^ — •= c 



2 7T 



dS dS dO t> • a ci 



.'. — — - ss — — . — - — = R n sin » cosec Ae c 

 dd de dS ° 



.\ con. surf. = / / R sin i cosec A { 1 — cos 2 A cos 2 ($ — «)} ^ o6 edHs. 



Now sin t cosec A { 1 — cos 2 A cos 2 ( <p — i) } I = { sin 2 * cosec 2 A 

 —cot 2 A sin 2 < cos 2 (<p — i) }* = {sin 2 < + sin 2 » cot 2 A 

 -cot 2 Asin 2 *cos* (<p — i)}*={sin 2 i + sin 2 »cot 2 A sin 2 ($ — »)}* 

 = {sin 2 » + c 2 sin 2 ($ — i)}* 

 .-. con. surf. =/TRo {sin 2 * + c 2 sin 2 ($-*)} V'rffl ds. 



Let s be taken to represent the value of s when = 0, 



• • * — *o • ' 



differentiating this value of s in respect to a given position of the 



ds c t 

 generating curve -% — = e 

 d s 



.-.conch. surf. =y^ «f sin 2 , + c 2 sin 2 (<j>-,)"| V'<Z0~ ds c 



= /7X { sin2 ■ + c2 sin2 (♦ - oV e2 c ' <* s . rf5 . 



Or integrating in respect to and observing that R , i, <p do 

 not involve 0, 



conch, surface = — — (s — 1 ) / R ] sin 2 « + c 2 sin 2 (4> — ») r d s 



(7.) 



where the integral / R -< sin 2 i + c 2 sin 2 (<p — ») > rfs re- 

 presents a constant determined by the geometrical form of the 

 generating curve, and its dimensions when 0=0. 



[The general form of the expression agrees with that given 

 in equation 15, p. 368 of a paper on the geometrical pro- 

 perties of turbinated and discoid Shells in the Phil. Trans., 

 part ii. 1838.] 



Phil. Mas. S. 3. Vol. 21. No. 138. Oct. 1842. X 



