1905-6.] Lord Kelvin on an Initiational Form. 
407 
Diagram No. 4 shows that, in the interval between its time and 
the time of No. 3, two zeros of the water-curve for ^-positive have 
come into existence. These and the corresponding zeros for 
x-negative are seen distinctly on the water- curve; and their 
indications for ^’-positive are marked by four crosses on the 
argument-curve. 
Diagram No. 5 shows that, between its time and that of ISTo. 4, 
twelve fresh zeros have come into existence on each side of 0 Z, 
one pair of which is indicated for example on the argument-curve 
by the parallel ^ 7 r. 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 x = 0 and x — ±\. 
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 ^fLr. 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 x= ± 1. 
§ 109. After the time l/J’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 (2^ + l)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 ^/ 7 r and SJtt, of diagrams 5 and 6, it has reached so close to x= 1 
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. 
§ 110. Each zero which originates according to an intersection 
on the outward side of the argument-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 originating 
according 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 
* If we continue the argument-curve to the side of the origin for 
ai-negative, we must include large negative values of i in (146) : but for 
simplicity we have confined the argument-curve to positive values of x. 
