110 TRANSFORMATION OF SURFACES BY BENDING. 



</' f/*z 



Differentiating these values of -7-5, &c., we shall obtain two values of , . , 



d*z 

 and of j-j-\, which being equated will give two equations of condition. 



Now if s be measured along a curve of the first system, and R be any 

 function of x and y, then 



dR dR dR . 



-j-f = -5- cos a+ ,- sui a. 

 as ax 1 1 ij 



, dR dRds' 



ana -, -, = -j-> -*, . 

 du as du 



We may also show that T - t = - , 



ds r 



da da d , ids' 



, ., da . da d , ids . ,\ 



and that cos a -5 sin a -y- = -y- log I -7, sin d> . 



dy dx ds = \du ^ 



By substituting these values in the equations thus obtained, they are 

 reduced to the two equations given at the end of (Art. 15). This method of 

 investigation introduces no difficulty except that of somewhat long equations, and 

 is therefore satisfactory as supplementary to the geometrical method given at 

 length. 



As an example of the method given in page (2), we may apply it to 

 the case of the surface whose equation is 



x \ + pLY = M'. 



This surface may be generated by the motion of a straight line whose 

 equation is of the form 



i 1, y = aemt (l + -} , 



t being the variable, by the change of which we pass from one position of the 

 line to another. This line always passes through the circle 



z = 0, x t + y > = a\ 

 and the straight lines z = c, x = 0, 

 and 2= c, y = 0, 

 which may therefore be taken as the directors of the surface. 



