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



and R s= 1*41 and taking the mean, we find the angle a = 83° 52'. 

 This example will illustrate the amount of coincidence that may 

 be hoped for from careful measurements. 



13. All these methods lead, of course, to erroneous results 

 when the shell has suffered distortion, except those derived 

 from measures taken along a single straight line. For ex- 



AF 

 example, the value derived from the ratio -^-^ ought to give 



HD UO ° 



the same result as ^^r (fig. 3), whatever the distortion: but these 



AE 



ratios will not be the square of yyj\. If the shell has been 



elongated in the direction AC in the ratio a : 1, we have 

 AE 2 . AE 





^2 HI) 2 - GC 



whence 



/AE /( 



VhdV] 



which gives a measure of the distortion. 



An example of this may be taken from specimens of 

 Goniatites, which are often distorted, in which state they 

 have been called Ellipsolitlies. Four whorl-breadths at right 

 angles were measured at 20, 17-J-, 14^, 12| lines respec- 

 tively, whence pp =1*378, ^ttt =1*372 (fig. 3), the differ- 

 ence being due to errors of observation on ill-preserved shells. 

 The mean may be taken as 1*375 for the true ratio. The 



values of the ratio derived from squaring the ratios tjrV "nn ? 



^cv respectively are 1-306, 1*456, 1*293. The first and third 



of these differ only from errors of observation : but their va- 

 riation from the true ratio is such as would be produced by 



distortion, the value of which is either ,\ / =*947 or 



A/ ^— t7^=*942 : that is, the diameter of the shell in the di- 

 V 1-456 



rection AC has been diminished by compression in the ratio 



•94. 



14. The value of the angle of retardation depends, of course, 



on the two points assumed. When the section is an ellipse, 



and the centre and extremity of major axis are compared, it 



depends on the length and position of the major axis. But in 



the more general case, whatever be the shape of the curve, 



especially if the whorls intersect, and in discoid shells, it is 



