24 
PEOFESSOE A. E. H. LOVE ON THE INTEGEATION OF THE 
The disturbance at any point, for which 2 > 2 ', is given by the following equations, 
in which /, m, n now denote the direction cosines of the line, of length i\ drawn from 
the point {x\ y', 2 ') to the point (x, y, 2 );— 
-^TTU = [[ dx'dy' ~ (cos k ( 2 ' — + r) + kv sin k {z — + r)] 
+ — „ fsin K (z — + r) — Kr cos k (z — ct ?•)} 
(m~ + n~) 
Kr” 
(3 — K'r) sin k [z — c/' + r) — Sk?’ cos k ( 2 ' — ct + r)} 
, (45). 
477 -r = j"! dx'dy sin «: ( 2 ' — + r) — ok)' cos k{z' — + '>')], • (- 16 ). 
47r?c = 
[ dx'dy' 
„ {cos K [z — ci + r) + KV sin k (z — + r )} 
In 
+ —y [(3 — K'r^) sin k {z — + r) — ’Sk)' cos k [z — -|- r)] 
K.') 
(47). 
Also 
477 /’ = II dx'dy' {(3 — kV^)cos k{z' + ct + r) + 3/c?’sin k{z' — + '^’)}> • (^^)- 
477p — [| dx'dy' K [sin k [z — ci + r) — kv cos k {z — + r)] 
2 
+ ^ [cos K ( 2 ' — + x) + KV sin k {z — ci + r) | 
— - {(3 — kV)cos/c( 2 '— ct + '?■)+ 3K>’sin k( 2 ' —ci + r)} , (49). 
477/? = |[ dx'dy' — ci + ?’) — ko' cos k {z — + r)} 
W 01 
+ -3 [(3 — kV") cos k (2' — Ct + 7 ’) + 3 k ?- sin k{z' — + r)} ( 50 ). 
At a great distance, the contribution of the element of area dS to {v, v, iv) is 
Kim . 
(n + nd + 7 d , — Im , — I — hi) — sin k (z' — ct r) ; . . , (51); 
Ttt )' 
the magnitude of this contribution is 
Kf/S . 
(1 + ?i) 7— sin K ( 2 ' — Ci! + x), .(52), 
47r7‘ 
and its direction is at right angles to that of and makes with the plane ( 2 , r) the 
same angle that this plane makes with the plane ( 2 , x). This direction is shown by 
