Lord Kelvin on an Initiational Form. 9 



Diagram No. 5 shows that, between its time and that of 

 No. 4, twelve fresh zeros have come into existence on each 

 side of OZ, one pair of which is indicated for example on the 

 argument-curve b}* the parallel \ 2 2 tt. Nine only out of all 

 the sixteen zeros on either side are perceptible on the water- 

 curve. The seven imperceptible zeros, on each side, all lie 

 between #=0 and x — +1. 



Diagram No. 6 shows that, between its time and that of 

 No. 5. forty-eight fresh zeros for ^-positive have come into 

 existence, one pair of which is indicated by the parallel -jf w. 

 Fourteen only out of all the sixty-four zeros on each side are 

 perceptible on the water-curve. Thirty-one of the fifty 

 imperceptible zeros on each side lie between x=0 and 



§ 109. After the time l/v/2, the zeros originate in pairs 

 on the two sides of the origin * (^-positive and ^-negative) : 

 those on the positive side by the two intersections of one of 

 the parallels corresponding to (2i + V)7r/'2 with the argument- 

 curve. The maximum of the argument-curve travels slowly 

 in the outward direction towards x=l as time advances to 

 infinity. At times 4 \/ir and 8*S7r, of diagrams 5 and 6, it 

 has reached so close to x = l that this point has been regarded 

 as the actual position of the maximum, both for the purpose 

 of drawing the curve, and for the determination of the total 

 number of zeros. 



<5 110. Each zero which oriolnates according to an inter- 

 section on the outward side of the arcmment-curve travels 

 outwards with increasing velocity to infinity, as time advances. 

 Each of the others of the pairs of zeros, that is to say, each 

 zero orioinatino- accordino- to an intersection on the inward 

 side of the argument-curve, travels very slowly inwards with 

 velocity diminishing to nothing as time advances to infinity. 

 Thus the motion of the water in the space between x = — 1 

 and x= + l becomes more and more nearly an increasing 

 number of inward travelling waves, with lengths slowly 

 diminishing to zero ; and, as we see by the exponential 

 factor in (144), with amplitudes and with slopes also slowly 

 diminishing to zero : as time advances to infinity. 



§ 111. The semi-period of one of these quasi standing 



Ct 2 



— IT O" 



waves is, as we find from (139), approximately equal to 



ozx 



when the time is so far advanced that \gt 2 is very great in 



* If we continue the argument-curve to the side of the origin for 

 ./■-negative, we must include large negative values of i in (14G) : but for 

 simplicity we have confined the argument-curve to positive values of x. 



