
Waves produced by any given Initiating Disturbance. 611 
by taking half sum of what/is written after it with +2, we 
haye, as a real solution of (4) for our problem 
—gt? 
p(x, 2, t nee a) Sari Mie i S/ (6) 
geez 
i [ ag 2 gh) ai 
a +2 pelea: — —z) sin, }€ “? 
pea Others gy tee 17 (7)", 
where p?=7" +2? 

e7 
=/- sin oe tape * | 
Se ee ee ee 
ee 
~ 
~ 
where 0=tan7! ie 
The sign of \/(p—z) changes when « passes through zero. 
Going back now to (5), and denoting by { rRD} the 
difference of its values for +z divided by 2u, we fee another 
solution of our problem essentially different from (6), as 
tollows 

1 ee 
Fe CT OE WL) aoe 
z+0x) 
baat) = {ED 
2 ? t)= L Say i 
1 oot Puc — 
= xv P+) sin fe —V/0—4) cost | 6) GLO, 
gt? v 
Dre git Tp 
=/ isin Ty +0—5 )e* ha Re ER (Gg 
§ 4. The annexed diagram, fig. 1, represents for t=0 the 
solutions ,6 and ,@ as functions of x, with z=1 for con- 
venience in the drawing. The formulas witch we find by 
taking t=0 in (7) x,/2 and 10x 4/2 are 
patti, gaviviet2)—a oy 
: / (£7 + 2”) oh Ke nf (i +27) va 
Before passing to the practical interpretation of our solutions, 
remark first that (12) contain full specifications of two dis- 
tinct initiating disturbances; in each of which ¢@ may be 
taken as a displacement-potential, or as a velocity-potential, 
or as a horizontal displacement-component or velocity, or as 
a vertical displacement-component or velocity. Thus we 
have really preparation for sz different cases of motion, of 
which we shall choose one, ~€=,/2 x (7), for detailed 
examination. 



# This solution was given in the Proc. Roy. Soc. Edinb. Jan. 7, 1887, 
and Phil. Mag. Feb. 1887. Itis quoted in App. C of “ Baltimore Lectures, " 
p- 931. 
ZT 2 
