8 Lord Kelvin on an Initiational Form. 



\/7r, -| s/tt, 4 s/tt, 8 >/ 7r. The formulas are simplified by taking 

 </ = 4. This is merely equivalent to taking as our unit of 

 length half the space descended in one second of time, by a 

 body falling from rest under the influence of gravity. For 

 simplification in the writing of formulas we take ~ = 1 for the 

 undisturbed level of the water-surface. The subsidiary 

 curves, explained in § 107 below, are called argument-curves, 

 as they represent the argument o£ the cosine in (144). 



§ 106. One exceedingly curious and very interesting feature 

 of these curves is the increasing number of values of x for 

 which the displacement is zero as time advances, and the 

 large figures, sixteen and sixty-four, which it reaches at the 

 times, 4=\/tt and 8^/tt, of the last two diagrams. These 

 zeros, for any value of t, are given by the equation 



A=(2i + l)7r/2 (146). 



§ 107. Notwithstanding the highly complicated character 

 of the function represented in (145), the zeros are easily 

 found by tracing an argument-curve, with A as ordinate, 

 and sb abscissa (as shown on the ^-positive halves of the six 

 diagrams on two different scales chosen merely for illustration, 

 not for measurement), and drawing parallels to the abscissa 

 line at distances from it representing — J-7T, —j>tt, \it, -|7r, 

 J-7T, &c. A parallel at distance —\ir is an asymptote to each 

 of the argument-curves, and is shown in diagrams 2, 3, 4, on 

 one scale of ordinates. The parallel corresponding to dis- 

 tance ^-7r is shown in the fifth and sixth diagrams, on the 

 smaller scale of ordinates used in their argument-curves. 



§ 108. The first diagram shows zeros at #=+^/3, of 

 which that at s i = — x/3 is marked 1. In the second dia- 

 gram the argument-curve indicates zeros for the — \ir and 

 —\ir parallels, which are seen distinctly on the water-curve. 

 The zero corresponding to the —\ir parallel was formed at 

 the origin at the time when \gf- was equal to z, that is, when 

 / was l/\/2, or *707. It is a coincidence of two zeros for 

 ^-positive and ./--negative. 



Diagram No. 3 shows that, shortly before its time, a 

 maximum has come into existence in the argument-curve, 

 which still indicates only two zeros. These are marked by 

 crosses. 



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 corre- 

 sponding zeros for ^-negative are seen distinctly on the 

 water-curve; and their indications for .^-positive are marked 

 by four crosses on the argument-curve. 



