432 Proceedings of Royal Society of Edinburgh. [sess. 
1 
CO 1 
( <*> ^ 
B 1 
; i j 
( m \ 
72 
__ V™ 2 . 1 . 3 1 
1 + 2 2 . 1 . J. o 1 
l J m ) 
1 
2 3 . 1 . 3 . 5 . Jm) 
4- etc. 
X_ K/W®_ 3 
(ufpY 
(200), 
^j2 2 2 13 2 ^ + 2 2 1 3 5 2 ^ 
(oijpy 7 
2M.3.5.7 C0S ¥ X+etC ' 
where, as in §§ 100-113 above, p = J(z 2 + x 2 ), and x = tan -1 (x/z). 
This series converges for every value of wjp however great. 
But for values of w x /p greater than 4, it diverges to large 
alternately positive and negative terms before it begins to con- 
verge. The largest value of wjp for which we have used it 
is o)Jp = 5 ’03, corresponding to a; = 8, and requiring, for the 
accuracy we desire, twenty-one terms of the series. But for this 
value of c ojp and for all larger values, we have used the 
ultimately divergent series (208), found in expressing analytic- 
ally, not merely for t = co as in (198), (199), (200), but for all 
positive values of t great and small, the growth to its final 
condition when t — oo , of the disturbance produced by our 
periodically varying application of pressure to the surface of the 
water initially (t = 0) at rest. The curve for ir in fig. 39 has 
been actually calculated by (200) for values of x up to 8, and by 
the ultimately divergent series for values of x from 5 to 10. The 
agreement between those of the values which were calculated 
both by (200) and by the ultimately divergent series (208), was 
quite satisfactory : so also was the agreement between values of 
Q found by quadratures for x = 1 and x = 8, with values found by 
(200) for &•= 1 and by (208) for a? = 8. It is also satisfactory that 
the values of P found by quadratures, for x=l, and x = 8, agreed 
well with their exact values given by (199), for t = go . 
§ 152. Going back now to the expressions (197) for P and Q, 
we see that, by an obvious analytical method of treatment, we 
can reduce them, and therefore (§ 150) all our other formulas, to 
expressions in terms of a function defined as follows : — 
dcr e “ 0-2 
( 201 ), 
