32 



Lord Kelvin on the 



^^^^[^-iiLafeT+^i 1 . 3.5(7^) 



2*. 1 



1 



©v'pcosf — 



(ft>\/p) 3 

 2.1.3 





X 



f>vA>) 5 o 



2MY3.5 COS 2* 



(200), 



_ _ ( w \/p) 



03 



1.3.5.7 e08 2 X + etc 



■] 



-lie 



re, as m 



100-113 above, 



but for all positive values of t 

 to its final condition when t — co 



p—\/(z 2 + ® 2 ), and % = tan- 1 ( t r/c). 



This series converges for every value of co^/p however 

 great. But for values of co^/p greater than 4, it diverges to 

 large alternately positive and negative terms before it begins 

 to converge. The largest value of co x /p for which we have 

 used it is 0)^/^ = 5*03, corresponding to a = 8, and requiring, 

 for the accuracy we desire, twenty-one terms of the series. 

 But for this value of coV p and for all larger values, we have 

 used the ultimately divergent series (208), found in expressing 

 analytically, not merely for t = co as in (198), (199), (200), 



great and small, the growth 

 , 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 quad- 

 ratures for x = l and a?=8, with values found by (200) for 

 .e — 1 and by (208) for a? =8. It is also satisfactory that the 

 values of P found by quadratures, for x — 1, and x=8 ao-reed 

 well with their exact values given by (199), for t = ao . 



§ 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 

 follows : — - r*a 



E(o-) = ! da-e-* 2 



a function well known to mathematicians 



as 



* The beautiful mathematical discovery 

 have been made by Euler about 1730. 



. . . (201), 



through the last 

 1 



(l(T€-<r 2 = - } sjn 



seems to 



