AND DEFORMATION OF A THIN ELASTIC SHELL. 511 



In the case of the sphere, this is 



- - i + ~ ( ~r + -7 ) where a is the radius, 

 P i P t a a \P\ Ptl 



or 



for any other inextensible surface this will not be the case. 



Now, suppose the surface extensible, and consider (a, ft) as two parameters defining 

 a point on the deformed surface ; in this view they will not be orthogonal parameters, 

 and we find 



i 



-(1+ 

 or 



F = -TT to the first order ; 

 *A 



so 



l-2<r, 1-2,7, 



~V~' ~V ' 



Again, 



A, 3f 

 - 1 + <r, 8. ' 



with similar expressions for m lt n t . 



To find /c' 2 , X'u K\ from the definitions in equations (5), we notice that the terms in 



"s/t\ rl / h \ 



ai \i + oj ' a0 \i + o-J 



will always be multiplied by terms of the form 



Now 



11 \ XT -r ^1 ^ 



, = K 2 - ,,) = h N s - ni M, == ^ j^ 



and similarly for wi 3 , 7t 3 ; thus, the differential coefficients of ^/(l + o-,), ^'(1 + <r) 

 will be multiplied by factors which vanish identically, viz., they are of the form 



, , 



a a ( a/ 8) "" a a (a/3) a a (/3) 



Hence, 



+ ff ^ ** 



V*. 



