TRANSFORMATION OF SURFACES BY BENDING. 467 



Supposing the surface to touch that plane at the origin, the length and tangential curvature 

 of the lines on the surface near the point of contact may be taken the same as those of their 

 projections on the plane, and any change of form of the surface due to bending will not alter 

 the form of the projected lines indefinitely near the point of contact. We may therefore 

 consider x as the only variable altered by bending ; but in order to apply our analysis with 

 facility, we may assume 



dx 



d l z 

 - — - = - PQ sin a cos a - PQ,' 1 sin 8 cos 8, 

 dxdy 



d 2 * 



— - PQcos'a + PQ 1 cos s 8. 



It will be seen that these values satisfy the condition last given. Near the origin we have 



= P s sin s (a-/3), 



_ d 2 * dV d'z 

 dx 1 dy* dwdy 



and q = Q -2 . 



Differentiating these values of — , &c, we shall obtain two values of and of 



da? dx'dy 



d 8 * 

 - — -— , which being equated will give two equations of condition. 

 vlcd dy 



Now if s' be measured along a curve of the first system, and R be any function of as and y, then 



dR dR dR . 



-7-r = -r- cos o + — sin a, 

 ds ax dy 



dR dR ds 



du ds' du' ' 



We may also shew that —7 = - . 



ds r 



, , da . da d , / ds . ,\ 



and that cos a — - sin a — = — log — -, sin . 

 dy dx ds \du I 



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 (446), we may apply it to the case of the surface 

 whose equation is 



\C-x) \C+!s) \c) 



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

 form 



w = a cos £ll — J , y = asin£(l+-j, 



