NOTE ON MR MONRO’S PAPER “ON FLEXURE OF SPACES.” 
[From the Proceedinqs of the London Mathematical Society, vol. ix. (1878), pp. 171, 172. 
Read June 13, 1878.] 
Consider an element of surface, surrounding a point P; the flexure of the 
element may be interfered with by the continuity round P, and it is on this account 
proper to regard the element as cut or slit along a radius drawn from P to the 
periphery of the element. This being understood, we have the well-known theorem 
that, considering in the neighbourhood of the origin elements of the surfaces 
z = \ (ax 2 + 2hxy + by 2 ), and z' = \ (a'x' 2 4- 2li'x'y' + b'y' 2 ), 
these will be applicable the one on the other, provided only ab — h 2 = a'b' — h' 2 . But 
in connexion with Mr Monro’s paper it is worth while to give the proof in detail. 
It is to be shown that 2, z denoting the above-mentioned functions of (x, y) and 
(pc', y') respectively, it is possible to find (for small values) x', y' functions of x, y 
such that identically 
dx' 2 + dy' 2 + dz 2 = dx 2 + dy 2 + dz 2 . 
The solution is taken to be x' = x + £, y' = y + y, where £, y denote cubic functions of 
x, y. We have then, attending only to the terms of an order not exceeding 3 in x, y, 
dx 2 + dy 2 + 2 (dxd% + dydy) + {(ax + h'y) dx + (h'x + b'y) dy} 2 
= dx 2 + dy 2 + {(ax + hy) dx + (hx + by) dy} 2 , 
so that the terms dx 2 + dy 2 disappear; and then writing