376 The Eev. J. H. Jellett on the Properties of Inextensible Surfaces. 

 The first of equations (C) gives us then at each of these points 



20 = 0. 



But since the position of the axis of a; is indeterminate, it follows from what has been said, 



that, on every section of the surface made by a plane passing through the axis of z, there 



will be at least two points, for which 



10 = 0. 



Hence it is plain that there will be on the surface one or more closed curves for which this 

 condition will hold. It will be sufficient to consider one of these curves, which, for the 

 sake of distinctness, we may call an equator. 



We have seen, p. 347, that lo must satisfy the equation 



d-w dho d-w „ 



dy'^ dxdy dx- 



or, as it may be otherwise written, 



d I dw dw\ d I dw dio\ „ 

 dy \ dy dx J dx\ dx dy j 



Multiply this equation by dxdy, and integrate it through the whole of either of the seg- 

 ments into which the surface is divided by the equator. We have then 



the single integrations being extended through the whole of the bounding curve. But since, 



for every point in this curve we have 



w = Q, 



if this equation be transformed according to the usual rule {Calculus of Variations, p. 218) 



it will become 



die' dw dio rfio^N 



dy'' dy dx dx- ) ' 



where ds is the element of the bounding curve, and 



/dw' dw'^ 

 \dx' dy- 



Now since in the class of surfaces which we are considering, 



rt - s"- > 0, 



it is easily seen that all the elements of the foregoing definite integral must have the same 

 sign. The total integral cannot therefore vanish unless each of its elements vanishes. Hence 

 it is plain that we must have at each point of the equator 



? = o. ? = o. 



dx dy 



