26 
PROFESSOR A. E. H. LOVE OX THE IXTE(4RATIOX OF THE 
such factors as + n~) and ■§(! + n) are of no importance in tlie ordinary appli¬ 
cations of expressions for the law of disturl^ance in secondary waves, and that, in 
fact, the enquiry after such a law involves a certain ambiguity. In the above deduc¬ 
tion of such a law, we have used the general formulse involving sources of two types ; 
if we could liave used fornndre involving sources of one type only, the result would 
probably have been different ; this is the origin of the ambiguity referred to by Lord 
Rayleigh. 
23. There is another difficulty attending the deduction of a law of disturbance in 
secondary waves from formulae applicable to the propagation of a system of plane 
waves, viz. : that integrals such as (45) taken over an infinite plane are not con¬ 
vergent. The disturbances in the secondary waves ought to combine to give rise to 
the disturliance actually propagated, or the result of the integration ought to be to 
reproduce the displacements in tlie primary wave. If we form such an integral as 
(45) for a portion of the plane {x, y'), and afterwards extend the boundaries of this 
portion indefinitely, we do not arrive at a definite limit. Let 0 be the point at which 
the distuiLance is to lie estimated. O' the foot of the perpendicular from O on the 
plane z = z (fig. 3), and let the portion of the plane lie bounded liy a circle, of 
o 
radius IT, with its centre at O'. We introduce plane polar coordinates /•', (/>, Avith 
origin at O', and put I = sinf^cosi/), m = sin 6'sin n = cos (9 ; then the value of 
the expression (46) for v is 
Jtt 
, r® , 7 , SUV 6 sin 6 cos cb 
d(h r d r -- 
^ J n /c r^ 
{(3 — K'l'-) sin K {z' — C? -f- r) — Skvcos k{z' — ct ;•)], 
and this \uxnishes identically, hoAvever great IT may be, on account of the symmetry 
of the circular boundary ; it would not have vanished if Ave had taken a boundary of 
a different shajie, or a circular boundary Avith its centre aAA'ay from O'. With the 
boundary chosen as above, Ave could shoAv that tv, f, h A’anish. To form tlie expression 
for n, Ave put 
+ (. - zf , =:= R'~ + (. - T)^ 
k{z' — - f - r) 
