170 EEPORT— 1898. 



complete curve soon becomes indistinguishable fi-om the two component 

 curves. 



On Several Outstanding Matters in Connection with the Chart. 



34. The long wave lines of the chart may be joined by a curved portion 

 to represent the complete long wave curve, which is the homologue of 



u-:=ak4- ^ . 



This may be obtained by using the method of the root of the sum of two 

 squares ; or it may be got by one setting of the slide rule from the short 

 wave gravity curve, this latter curve being the homologue of 



c c\2 



(--!») 



= [aX+y^ 

 c 



In this case the A being read directly from the chart no thought is 

 necessary in the operation, the lengths being simply transferred by the 

 proportional compasses. On the chart when p, s, p, E have the quoted 

 values and e=100 the complete homologue to the original equation is 

 practically coincident with the long wave curve in the curved portion of 

 the latter. 



35. It is interesting to compare one of the homologues of the original 

 equation with the wave and ripple curve of Mr. Boys. In the latter on 

 the left we have the capillary ripples where the curve is straight, then the 

 curved portion as far as the point of minimum velocity represents ripples, 

 in the propagation of which gravity has an increasing influence : this is 

 followed by a curved portion in which the surface tension eflfect is waning, 

 and the gravity influence waxing ; finally we have the straight portion 

 which represents waves in which the influence of capillarity is negligible. 



The shape of the curve on the chart varies as the quantities ff, e, E, p, 

 and s alter, as has already been seen. The elasticity and gravity branches 

 always become coincident with the short wave elasticity and long wave 

 gravity branches as we pass to the left and right respectively through a 

 sufiicient range. 



Consider now the complete curve on the chart representing approxi- 

 mately the facts when the solid is ice and the liquid water and e=l. 

 Waves whose length is less than about 420 correspond to ripples ; when 

 the wave length falls below about 80 we have the analogues of capillary 

 ripples. 



36. The value of X for which u is a minimum is a solution of the 

 equation 



X^' + 2c\^-^lx-^=o 

 a a 



obtained by diflerentiating the original equation and putting ^- =o. The 



ctX 



ordinary practical way of solving this equation (by numerical compu- 

 tation) is by Horner's method. It has only one real root which is approxi- 

 mately 419 with e=l. This root is at once given approximately by the 



