1905-6.] Lord Kelvin on the Growth of a Train of Waves. 433 
a function well known to mathematicians* through the last 
hundred and fifty or two hundred years, in the mathematical 
theory of Astronomical Refraction, and in the theory of Probabili- 
ties. I have taken E as an abbreviation of Glaisher’s f notation 
“Erfc,” signifying what he calls “Error Function Complement,” 
which he uses in connection with his name “Error Function,” 
defined by 
Erf (<r)= f d!<r«-^=y Jir - Eric(<r) . . (202). 
ho- 
using the imaginary expression for <£ in § 137, we find 
P = { ES } f f e dq [ « " G"' “ ‘ 2 + e ~ | (203); 
q = {RS}\f~e^ffq[e-( Vmt -^f-e-( Vmt+ ^y~\ ( 204 ), 
where m= ~g/(z + lx), as in § 151. 
Taking advantage now of the notation (201), we reduce these 
two expressions to the following : — 
— lifi 
{RS> \/7 € ‘ M K Jmt ~ ‘2 fn) + E ( + ‘2 i). 
(205); 
Q= j elZ K - E ( 2E fe)" 
(206). 
§ 153. Remark first in passing that, when Jmt is infinitely 
great in comparison with <o/2 Jm, these two expressions agree 
with the expressions, (198), for P and Q with £=oc, which we 
used in connection with the explanation of fig. 39. 
§ 154. And now, with a view to finding P and Q for any chosen 
values of x, z , t, we have the following known series | : — 
* The beautiful mathematical 
discovery, / 
Jn 
dart- 0-2 1 
seems to have 
been made by Euler about 1730. 
t Phil. Mag., October 1871. 
$ See Glaisher, “On a Class of Definite Integrals,” Phil. Mag., October 
2 r 
1871 ; and Burgess, “ On the Definite Integral —r I 
dt, v Trans. Roy. Soc. 
Edin., 1898. 
PROC. ROY. SOC. EDIN. — VOL. XXVI. 
28 
