258 Rev. J. F. Blake on the Measurement of the 



circular. Hero 



tan co 2 = 



4 



38 = 



2 



"19 



» 



tan &>! = 



8 

 : 5 X 



4 



38 = 



16 

 = 95 



ban co\ = 



: tan 



&/ 2 = 



1 



: 8* 



also 



In this case the two above expressions become respectively 

 *0071 and '0048. Hence the tapering in the compressed shell 

 compared is more rapid. If a conical shell be perfectly flat- 

 tened and its transverse ornaments become circular arcs, the 

 length of any semicircumference = itci gives the original dia- 

 meter of the shell when circular. 



18. In the case of a curved shell of elliptical section being 

 compressed perpendicularly to its median plane, we may 

 assume that the area remains constant, and find the effect on 

 the angle /3. 



If a = e ecota cr, the element of area 



= e 2ecot «(r*/ 1 -e 2 sin 2 </> #(1— <rsin <j>)d0 



when <j) is measured from the end of the minor axis towards 

 the pole, and the same with 1 + a sin cf> instead of 1 — a sin cf> 

 when measured away from the pole. Integrating with respect 

 to 6 between the limits 6 and — co , the area may be written 



at a/* 



2 cot ~ J ^1 — e 2 sin 2 <£(l±o-sin<£)£Z(/>. 



Now, denoting I ^ \ — e 2 sin 2 <f> d$ by E(e<£) as usual, and 

 Jo 



f Vl-e 2 sin 2 <£ sin cf> d(f> by M(e</>), 

 Jo 



^ a =lS{ E (4) +E(e ^ +ff K e i)- <rM ^}- 



Hence, a remaining unaltered, the condition that a compressed 

 and uncompressed shell may belong to the same species is 



°" E ( e I) + °" E ( e *) + °" 2M ( e f )-°" 2 (Me<W = { ^(e'f) 

 + c7 / E(e / </>0 + o- /2 M^e / |)-cr /2 M(e / (/) / )}^ cot ^'- e) . (1) 



