1078 Proceedings of Royal Society of Edinburgh. 
b 
d(x, y) = 
little from it, we take + q 2 instead of the -qf of (111), and 
(d 2 u/dx]/ 2 ) 2 instead of the - (d 2 u/dx{/ 2 ) 1 of (115); because u x is the 
maximum and u 2 the minimum. Calling k v k 2 the values of k 
corresponding to x /q, x}/ 2 , and using these expressions properly in 
(113), we find, for the depression of the water at ( x , y), 
4 &7T 3/2 
S 1 COS 2 xf / 1 
f dq x sin (a x - q-f) + — ^ [ dq 2 sin (a 2 + ? 2 2 )1 . ( 1 1 7). 
J Kin COS Xj/nJ j 
/3 2 cos 2 xf/ 2 
The limits oo , - oo are assigned to the integrations relatively 
to q 1 and q 2 because the greatness of r/A in (115) and correspond- 
ing formula relative to xf/ 2 , makes q 1 and q 2 each very great, 
(positive or negative,) for moderate properly small positive or 
negative values of xf/-xf/ 1 and xj/-xf/ 2 . Now as discovered by 
Euler or Laplace (see Gregory’s Examples , p. 479), we have 
r oo /*oo 
I dq sin q 2 = I dq cos q 2 = Jtt/2, and using these in (117) we 
find 
d(x !/) - 2 ^ 27r ^ f ^i( sin a i ~ cos a i) , *g( sin a 2 + cos a 2) 1 ( 118 \ 
h L cos 2 «/q fi 2 cos 2 xf/ 2 J 
Substituting for a p a 2 values by (115) we find 
4t r 2 b 
. 2t r( 
sm ( ru j — 
A N 
) + ^2 
• 27 T[ 
sin 
A 
COS 2 x [/ 1 
A \ 
8 / 
> fi 2 cos 2 xJ / 2 
A\ 
(118)'. 
§ 85. To determine the quantities denoted by /3 V j3 2 in (116) 
. . . (118)', we write (112) as follows : — 
ru = (x + yt) Jl+t 2 , where t = tan xf/ . . .(119). 
Hence, by differentiation on the supposition of x, y , r constant, we 
find 
y{ i + 2* 2 )] Ji 
du 
r # = L' 
xt + 
+ t 2 
d 2 uV 
x(l + 2< 2 ) + yt(5 + 6i! 2 ) L/l + t 
. ( 120 ). 
• ( 121 ). 
By (120) we find for t m , which makes u a maximum or minimum, 
xt m + y( l + 2^) = 0 (122); 
a quadratic equation which, when (y/x) 2 <^, has real roots as 
follows, — 
~Ty V[(f,)‘- i]' V[(5)’- ?] < 123 >’ 
