Colloquium publications.

130

THE CAMBRIDGE COLLOQUIUM.

with

Pij = Oij,

Oall 9 dai2
(55) x+2 ai,

oa12 a 22
2 = 2 + 2 -a2,
ax oy

2an ll 02a2 a22 a2al2 Oai 0a2
P = + 2 y - 2 - - - a.
dx2 + dxdy 0y2 dx dy
The formulae for M(-r) and its coefficients are the well-known
ones for differential equations, except that attention is paid to
the order of the factors, to avoid the necessity of a hypothesis
of permutability.
If we write

Si =(~a11nax+ a12~ d)o~ o~

(56)

( dX 0+ ay12 J
/ 9Oaai 0 a12
+r al — d-x- dy )i'
( a2 - d-y ' 22)
0aal2 0a22
+ n a2 - ax -- y

Ox ( a2 -

we have Lagrange's identity:
OS1 0S2
(57) qL(t)- M()t.= Ox + Oy
and Green's theorem:

(58) fS [L() - M(r7)~]dxdy = f [Si cos x, n + S2 cos y, n]ds.
The equivalent of this theorem was given first by Volterra for
the integro-differential equation appearing as a generalization of
Laplace's equation in three dimensions. In two dimensions,
the equation is obtained by writing

a11 = 1 + jAll(rs),

a22 = 1 + jA22(rs),

(59)

a12 = a1 = a2 = a = 0.