260 Prof. Challis on Hydrodynamics. 



tions, and is therefore greater for a given longitudinal velocity 

 than it would be if the effect of the transverse vibrations were 

 neutralized, as is the case in the composite motion. In order to 

 find the relation between V 1 and a x , recourse must be had to 

 the abstract relation between velocity and condensation uni- 

 formly propagated in parallel lines, which is investigated in 

 Part I. (p. 218). It is there shown that to the first approxi- 

 mation Y x = Kacr 1 . Hence, if 



o_ 

 Y 1 = X . m sin — (z—mt + c) —f{z — Kat -f Cj), 



A/ ■ 



we shall have a } = — ./(z — rnt + c-^. On the principle of the 



coexistence of small vibrations, contemporaneously with the pro- 

 pagation of Vj and c x in the positive direction, there may be 

 propagated in the contrary direction the velocity V 2 and conden- 

 sation cr % such that V 2 =— /eao-Q=¥{z + mt + c^), Hence if 

 VrsVj + Vg, and o- = <r 1 -{-<r 2 , we have 



Y = f(z— /cat + c x ) +¥(?-{• /cat -h %), 



tcacr =-f(z — Kat + c x ) — F {z + Kat + c 2 ) . 



It follows from these two equations that 



2 L da dV . 



Ka -T z + lt=° W 



This differential equation takes account of the composite charac- 

 ter of the motion, and, as far as regards the first order of approxi- 

 mation, is of perfectly general application. For although the 

 axes of the component motions were supposed in the investiga- 

 tion to be rectilinear and parallel, since this is a differential 

 equation, by a well-known principle of analytical reasoning, it is 

 ultimately applicable to curvilinear and inclined lines of motion, 

 provided they are everywhere inclined to each other by indefi- 

 nitely small angles. Hence dz may be assumed to be the same 

 as ds, the increment of a line of motion, and the equation will 

 thus agree in form with the general dynamical equation of the 

 first order applicable to uncompounded motion, only having K?a 2 

 in the place of a 2 . 



Conceive now waves to impinge on a sphere at rest, and let 

 the centre of the sphere be the origin of the polar coordinates r 

 and 6, the angle 6 being measured from the part of the axis 

 directed towards the origin of the waves. Also let U and W 

 be the resolved parts of the velocity respectively along and per- 

 pendicular to the radius vector, so that V 2 =U 2 + W 2 . Conse- 

 quently 



