IOC TRANSFORMATION OF SURFACES BY BENDING. 



16. To obtain the condition of Instantaneous lines of bending. 



We have now obtained the values of the differential coefficients of q with 

 respect to each of the variables u, '. 



From the equation 



we might find an equation which would give certain conditions of lines of 

 bending. These conditions however would be equivalent to those which we have 

 already assumed when we drew the systems of lines so as to be conjugate to 

 each other. 



To find the true conditions of bending we must suppose the form of the 

 surface to vary continuously, so as to depend on some variable t which we 

 may call the time. 



Of the different quantities which enter into our equations, none are changed 

 by the operation of bending except q, so that in differentiating with respect 

 to t all the rest may be considered constant, q being the only variable. 



Differentiating the equations of last article with respect to t, we obtain 

 d* n , 2 ds 1 d 



eP /, 2 ds 1 1 d 



^(log2) = 



Whence 



Zdsl d x d , , 2 cfcs 1 d 



rd^^ + rTu 2E? 



and 



d /2 ds' 1 \ 2 ds' 1 d , \ 1 d ., 2 ds' 1 1 d 



^ VW ^/ ~ rWshT^ 10g ? J ? cfc (1 g ^ 



.. 



two independent values of the same quantity, whence the required conditions 

 may be obtained. 



