270 The Rev. J. Challis o?i ihe analytical Determination 



state of the fluid ; and whether, previously to their application 

 at any subsequent epoch, the law of transmission must not be 

 first deduced by means of the quantities which the arbitrary 

 functions involve. 



The means of meeting the difficulty stated above will, I 

 conceive, be furnished by obtaining in an independent man- 

 ner a general expression for the velocity of the propagation 



ds 

 of motion, analogous to the expression -y- for the velocity of 



motion itself. The employment of such an expression for the 

 determination of the velocity of propagation is the principal 

 feature of the method I am about to propose*. 



The following reasoning, which is in part the same as that 

 in p. 253 of Professor Airy's Mathematical Tracts (2iid edit.), 

 will answer the purpose we have in view. Let y = ^ {oc, t), 

 as it must be some function of x and t. Suppose any given 

 value of the ordinate to be carried through space with the 

 velocity v during the small time t. In general v will be a 

 function of x and t; but we may suppose it constant during 

 the time t, and for a small portion 8 or of the axis of abscissae, 

 because by this supposition only quantities of the order t\ 

 tSot, and above, will be neglected. Hence for the same small 

 interval of time, the function 4^, as far as it relates to the small 

 portion Ix of the line of abscissae, may be considered invaria- 

 ble. Consequently, 



y = (P(j:,t) = (^(jc + vr,t + T) 



Hence when t is indefinitely diminished, -f^ v + -~ = 0, 



ax a c 



and u = r— . This is the formula it was required to ob- 



dy 



dx 

 tain. It is of very general application, and will serve either 

 to find V in terms of x and ^, when y is given a function of 

 these variables, or when *o is given in like manner, to find y 

 by integration. If the formula be applied to such an equa- 

 tion as ^ = — 5 the X in the denominator may be 



• I have obtained a like formula in the Philosophical Magazine and 

 Annals, N.S , for May 1830, (vol. vii. p. 325,) with particular reference to 

 the propagation of motion in a compressible elastic fluid. 



