ARISING FROM THE LOAD OF NEIGHBOURING OCEANIC TIDES. 
43 
At a distance from the edge of the loaded circle, l is a fairly small quantity and 
consequently the terms after second or third may be dispensed with. The q series of 
the functions needed here are as follows :— 
T/i + ejWi — 
«i 
mil+2 2 q2n , 
4 -Kf'f/ 
Ilf] »•*«>). 
„ _ i f y",(o) a", (o)i. 
‘“V'' 1 8„,U',(o) a 3 (o)/’ 
b 2 (0) = 2g : (l +g 2 + g 6 + g 12 +...), 
b'" (0) = -27T 3 ^(l-3 3 g 3 +5Y-7V 3 +...) 5 
b^O) = 27rg i (l —3g 2 +5g 6 —7g 12 +...), 
b" 2 (0) = — 2 tt Y (1 + 3 2 g 2 + 5 2 g 6 + 7 2 g 12 + ...). 
At the point near the edge of the loaded circle, the above expansions cease to be 
applicable. For this case, our object will be accomplished by using the quantity q u 
instead of g, which is defined by 
q x is calculated from 
in which 
{e x -e^f — (e 2 —e 3 )* (?■ + «)* — (2 {ar) h f 
(ei-c 3 ) : + {e 2 -e z ) 1 (r + af- + (2 (arff 
For example, to calculate ( ‘Y) near the edge, we proceed like this :— 
\ or /o 
By the aid of the relation 
(o) b " 0 b" 2 , b " 3 
^(o) b 0 b 2 1 b 3 ’ 
* 
the function ^e l w 1 — >] X may be transformed into 
(i e i w i— m) ~ 
b 2 2 (0 | t) 
f b" 0 (o 1 t) 
l bo (o | t) 
b'h (0 I t) I _ 
b 3 (0 j r) J 
Making use of the transformation formulae of Theta-functions it will be easily 
shown that 
avoir) 
b 0 (0 | r) 
S",(0\t) 
b 3 (0 |t) 
2'Z'7TT 1 + Tj 
y'aoL,) 
(II ' T, ) 
) 
2<7TT 1 + Ty 
b" 3 (0 |Tj) 
b 3 (0 | Ti) 
VOL. CCXVI1.-A. 
H 
