The Kev. J. H. Jellett on the Properties of Inextensible Surfaces. 375 



NOTE. 



Since the foregoing sheets were printed, I have arrived at the following theorem, which is 

 of some interest, as connected with the class of surfaces which we have been examining : 



If a dosed oval surface heperfecUy inextensible, it is also perfectly rigid. 



To prove this, let us denote, as before, by ex, Sy, Sz, the resolved displacements of any 

 point on the surface. Let S!x,B'y,Sz be its most general displacements considered as a rigid 

 body ; then it is known that 



S!x = a + Cy - Bz, 



S'y = b + Az - Cx, 

 ^z = c + Bx - Ay, 



a, b, c, A, B, C, being constants. Now if we form a third system — 



Ax = Sx + S'x, 

 Ay = By+ ^y, 

 Az =Sz + 8'z, 



it is plain that Ax, Ay, Az will satisfy the conditions of the problem contained in equations 

 (B) or (C). Moreover, if x,yiZi, XiyiZi be two given points on the surface, the constants 

 a, b, c, A, B, C, can always be so determined as to satisfy the equations 



Axi = 0, Azi = 0, 

 Ax2 = 0, AZi = 0, 



without in any way limiting the generality of the displacements Bx, By, Bz. Suppose now 

 that we assume, as in p. 346, 



u = Ax + pAz, w = Az, 



it is plain that u, w will satisfy the first of equations (C), and will vanish at the two points 

 XiyiZi, x.^.Z2. Let these points be P, Q, and suppose, to fix our ideas, that the axis of z 

 passes through them. The plane of xz will then intersect the surface in a closed curve, 

 PRQS, passing through these points. Now since u vanishes at the points P, Q, if we 

 trace its values in passing along the curve PRQS, we shall find a maximum value (dis- 

 regarding its sign) somewhere between P and Q as at R, and again somewhere between 

 Q and P as at S. We have, therefore, for each of the points R, S, 



since the equation of the curve PR QS is 



dy = 



