92 Prof. Challis on the Principles of Hydrodynamics. 



finitely from the axis, it is possible that by a combination of 

 different sets of vibrations with different values of the constants, 

 the distant motions and condensations may be destroyed, and 

 an initial disturbance of finite extent be composed of normal 

 vibrations. 



In the other class of disturbances the velocity is given at given 

 positions in successive instants, and the condensation is to be 

 inferred from the given circumstances of the disturbance. The 

 fluid may be conceived to be set in motion by the surface of a 

 solid, and to be either unlimited in extent, or to be confined 

 within solid boundaries. The motion in such cases is constrained, 

 and may be conceived to be composed of parts of the normal 

 motions for which m, \, and c are constant only for an indefi- 

 nitely small time and through an indefinitely small space. 



In the application of these principles to the undulatory theory 

 of light, the initial disturbances appear to be of the first class, 

 and not to be immediately or necessarily due to the motion of a 

 solid ; the motion contiguous to an axis of propagation, which 

 is analytically distinguished by satisfying exactly the integra- 

 bility of udx-\-vdy-\-wdz, is the exponent of a ray of light; and 

 the direct vibrations, and the parts of the transverse vibrations 

 remote from the axis, must be supposed to be incapable of pro- 

 ducing the sensation of light. 



Before proceeding to apply the above results to an instance of 

 arbitrary disturbance of the fluid, the circumstances of uniform 

 rectilinear propagation in cases of constrained motion are first to 

 be investigate^i. 



Proposition XII. To determine the relation between the velo- 

 city and the density, when uniform propagation takes place in a 

 straight tube whose transverse section is of arbitrary magnitude, 

 but everywhere indefinitely small. 



Let V and p be the velocity and density of the fluid which 

 passes the transverse section m at the distance z from the origin 

 at the time /, and let V and p' be the velocity and density of the 

 fluid which at the same instant passes the transverse section m' 

 situated in advance of the other by the interval hz. The incre- 

 ment of fluid between the two sections in the interval from 

 , ^t ht . 



'~ 2 ^"^ ^ "^ 2 '^ mpYBt-m'p'Y'St, 



because the changes of pY and p'V in the small interval Bt may 

 be supposed to be proportional to the time. Let the above 

 increment become equal to the excess of the quantity of fluid at 



the time /— — in the element of length Bz terminating at the 



section m, above the quantity of fluid at the same time in the 



