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



IT 



When the whorls do not intersect, we must put $ = 77 in (1) 



and cancel the fractions in (6): the rest are not required. 

 The only important application of these conditions is when a 

 shell has been completely flattened in shaly beds. In any case 

 crumplings of the surface may be induced by resistance to 

 lateral expansion, when the expression for the area of the com- 

 pressed shell will come out less than it ought ; but if it comes 

 out greater, the two shells compared could not have had the 

 same original form. An example may be take in Ammonites 

 planorbis, which, when uncompressed, has been called Ammo- 

 nites erngatus. In an example of the latter we find 11=1*33, 

 \=*431; whence by (5) <r=*449; and by (2) sin</> = *838. 

 The last whorl on which these measures were taken was 10" 83 

 millims. in breadth; .*. 10*833 — <2=*838a, or a = 5*894. The 

 thickness of the same whorl was 8*33 millims., whence e = *7065. 

 The elliptic integrals are best observed directly by measure- 

 ment of the arc ; and if the values differ slightly from the 

 theoretical values, the correction has probably to do with the 

 deviation of the form from a perfect ellipse. In this case 



2a{E(e / |)+E(e / ^)} 



= the circumference of the whorl between its intersections 

 with the inner whorl = 26 millims, whence 



E(e,|)+E(e'<£) = 2*205. 



Also, on substitution, 



M(e,</>) = *408, and M/o^Wsil. 



Hence the first side of equation (1) becomes 1*071. In the 

 flattened shell as seen in the type, R = l*33 and X/ = *384, 

 whence o- / = *49 and a' sin ft = '£28. Also, e' being unity, 



E (V, |) + E(e', <f>) = 1 + sin^, and M(ey) =?^-', 



and 



M 



(e',f)=i 



Hence the second side of the equation reduces to 



1-038. #?-*)«**. 

 Now we have seen that 



E(e,|)+E(e,</>) = 2-205; 



