332 
NOTE ON MR MONRO’S PAPER “ ON FLEXURE OF SPACES. 
[691 
the equation will be satisfied identically as regards dx, dy if only 
2 ^ = {ax + hyf — {ax + h'yf, 
^ -1- ^ = {ax + hy) {hx + by) — {ax + h'y) {Iix + b'y), 
2 ^ = {hx + by) 2 - {h'x + b'y) 2 . 
Calling the terms on the right-hand side 221, 23, 2(£ respectively, we have 
that is, 
dm_d^ d ^ = 0 
dy 2 dxdy dx 2 ’ 
{h 2 - h' 2 ) + {h 2 - K 2 ) - {{ab + h 2 ) - {ab' + h' 2 )} = 0, 
or, what is the same thing, 
a'b' — h' 2 = ab — h 2 , 
a relation which must exist between the constants {a, b, h) and {a', b', h'). 
It is easy to find the actual values of £, 77; viz. these are 
£ = £ (a 2 — a' 2 ) a? + {ah — a'h') x 2 y + (A 2 — h' 2 ) x 2 y + £ {bh — b'h') y 3 , 
V = i {ab — a'h') « 3 + 2 {b 2 — h' 2 ) x 2 y + ^ {bh — b'h') x 2 y + ^{b 2 — b' 2 ) y 3 , 
or, what is the same thing, we have 
£_ 1 _ 1 
S-^dcc’ V ~^dj > 
where 
Q = {a 2 — a' 2 )x* + 4 {ah — ah')afy + 6 {h 2 — h' 2 )x 2 y 2 + 4(bh — b'h')xy 3 + (b 2 — b' 2 ) y 4 , 
= {ax 2 + Zhxy + by 2 ) 2 — {ax 2 + 2 h’xy + b'y 2 ) 2 = 4<{z 2 — z' 2 ), 
in virtue of the relation ab — h 2 = a'b' — h' 2 . The resulting values x' = x + £, y = y + 77 
are obviously the first terms of two series which, if continued, would contain higher 
powers of {x, y).