Growth of a Train of Waves. 33 



hundred and fifty or two hundred years, in the mathematical 

 theory of Astronomical Refraction, and in the theory of 

 Probabilities. I have taken E as an abbreviation of Glaisher's * 

 notation " Erfc/' signifying what he calls " Error Function 

 Complement.''' which he uses in connection with his name 

 "Error Function/'' denned by 



Erf(«r)=i </a-e--=^/7r-Er£c(c7) . (202). 

 Using the imaginary expression for <b in § 137, ye find 



V 



q = {RS}^/^'e^(^^ (204), 



{BS}a/ — e 4 * J dqy { W»J +e v -V-M (203); 



W2 f»$ 



where m= -j o'(~+cx), as in § 151. 



Taking advantage now of the notation (201), we reduce 

 these two expressions to the following : — 



+ B( v /- t + ^5_)] (205): 



- E (v^ + '2^> 2B ( l 2 



0) 



(206) 



§ 153. Remark first in passing that, when \/mt is infinitely 

 great in comparison with wj'2\/m, these two expressions 

 agree with the expressions, (198^ for P and Q with £ = oo, 

 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 



■W--^QT-GJ? + ' • ' < 207 >< 



* Phil. Mag., October 1871. 



f See Glaisher, " On a Class of Definite Integrals," Phil. Mag., October 



] 871 : and Bunress, " On the Definite Integral *-j-\ e-* 2 dt" Trans. Roy 

 Soc. Edin., 1898. ' V77 Jo 



Phil. Mag. S. 6. Vol. 13. No. 73. Jan. 1907. D 



