86 Prof. Challis on the Principles of Hydrodynamics. 



pend upon the size of the drops of rain, although the width of 

 the primary bow is independent of this. The larger the drops 

 of rain, the closer arc the bows ; and therefore under some cir- 

 cumstances the bows may be so close together, that at points 

 within the primary bow, the mixture of colours may be such as 

 to make the bows invisible. This may account for the fact, that 

 the supernumerary bows are not always seen accompanying the 

 primaiy bows. 



N. W. Provinces, India, 

 Sept. 18, 1852. 



XVI. On the Principles of Hydrodynamics. By the Rev. J. 

 Challis, M.A.y F.R.S., FM.A.S., Plumian Professor of 

 Astronomy and Eocperimental Philosophy in the University of 

 Cambridge. 



[Concluded from vol. iv. p. 450.] 



THE antecedent investigation of the free motion of a com- 

 pressible fluid, led to the general law that the motion is 

 symmetrically disposed about a rectilinear axis, along which a 

 given state of the fluid as to density and velocity is propagated 

 at a certain constant rate. An expression for the rate of propa- 

 gation was obtained involving an unknown constant 6^, the value 

 of which it is necessary to ascertain in order to calculate nume- 

 rically the velocity of sound. I proceed now to show how that 

 constant may be determined, and a solution be given to the fol- 

 lowing proposition. 



Proposition XI. It is required to calculate numerically the 

 rate of the propagation of motion in a given compressible fluid. 



I have given a solution of this important question in the 

 Philosophical Magazine for February 1849 (vol. xxxiv. p. 97), 

 but as the reasoning there is not adequately exhibited, I propose 

 to repeat it*now in more detail. It has been shown that the free 

 vibratory motion of a compressible fluid is symmetrical with 

 respect to a rectilinear axis, and may be resolved into motions 

 parallel and transverse to this axis. 



The component parallel to the axis s=/-^. 



The component transverse to the axis = (/> -j-. 



The condensation at any point = — ^g* -^, 

 The value of the quantity / is given by the series 



J — I ei + j2 2* 1^.2^.3'^ +a;c., 



