TRANSFORMATION OF SURFACES BY BENDING. 109 



Let a tangent plane move along any line of the first system, then if the line 

 of ultimate intersection of this plane with itself be always a tangent to some line 

 of the second system, the second system is said to be conjugate to the first. 



It is easy to show that the first system is also conjugate to the second. 



Let the system of curves be projected on the plane of xy, and at the point 

 (x, y) let a be the angle which a projected curve of the first system makes with 

 the axis of x, and )8 the angle which the projected curve of the second system 

 which intersects it at that point makes with the same axis. Then the condition 

 of the systems being conjugate will be found to be 



d'z d'z 







a and /8 being known as functions of x and y, we may determine the nature 

 of the curves projected on the plane of xy. 



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 z as the 

 only variable altered by bending; but in order to apply our analysis with facility, 

 we may assume 



^ = PQ sin 1 a + PQ- 1 sin' & 



= PQ sin a cos a PQ~ ' sin /3 cos /3, 



dxdy 



cPz 

 3? = 



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

 origin we have . 



d'z d'z d'z * 



^_ 



* dx* dy* dxdy 

 and q = Q~*. 



= Psin 2 (a-/3), 



