10 Lord Kelvin on an Initiational Form. 



comparison with p. Thus we see that the period is infinite 

 at the origin. This agrees with the history of the whole 

 motion at the origin, which, as we see by potting # = ift 

 (139), with c = l and # — 4, is expressed by the formula 



-f=i(l-2^e-* .... (147). 



The motion of the water in the space between #=— 1 and 

 #=+1 is of a very peculiar and interesting character. 

 Towards a full understanding of it, it may be convenient to 

 study the simplified approximate solution 



£== — , cos ( V — -» tan -1 x je p- . (118) - 



a-'*- \ p~ 2 / 



which the realised part of (139) gives when ^gt- is very large 

 in comparison with p. 



§ 112. The outward travelling zeros on the two sides, 

 beyond the distances +1 from the origin, divide the water 

 into consecutive parts, in each of which it is wholly elevated 

 or depressed. These parts we may call half-waves. They 

 travel outwards with ever-increasing length and propagational 

 velocity. Each of the half-waves developed after t= s/m, as 

 it travels outward, increases at first to a maximum elevation 

 or maximum depression, and after that diminishes to zero as 

 time advances to infinity. 



§ 113. It is interesting to trace the progress of each of the 

 zeros in the intervals between the times of our six diagrams. 

 This is facilitated by the numbers marked on several of the 

 zeros in the different diagrams. Thus, confining our attention 

 to the left-hand side of fig. 34, we see in diagram 1 a single 

 zero numbered 1. The future zeros are to be numbered in 

 the order of their coming into existence, 2 ; 3, 3 ; 4, 4 ; . . .; 

 10, 10; . . .; 33, 33; ... all in pairs after zero 2. Thus 

 diagram 2 shows zero 1 considerably advanced leftwards 

 (that is, outwards) ; and zero 2 beginning its outward 

 progress. Diagram 3 shows zeros 1 and 2 each advanced 

 farther outwards, 1 farther than 2. Diagram 4 shows all the 

 zeros which have come into existence at time -fv^ 77 "- These 

 are zeros 1 and 2, both farther outwards than at time \/ir, 

 and a pair, 3, 3, which have come into existence shortly 

 before the time § <Jir. The outer of these two travels out- 

 wards and the inner inw-ards. Some time later 4, 4 come 

 into existence between 3 and 3 : later still 5, 5 come into 

 existence between 4 and 4. 



In diagram 5, zero 1 has passed out of range leftwards : 



