586 Professor CHALLIS, ON A MATHEMATICAL THEORY 



since ^ = <pf= - arjsdt, it follows that /"jy = " «'«- 3"^ 57 = " o^-^' ^^"" '"'''''- 

 tuting in the equation (4), we obtain, 



de d^ f W df)^ ' 



Now the nature of the question under consideration requires that this equation should be linear. 

 Let therefore the coefficient of (p be equal to a constant - 6^. According to this supposition 

 (p may be a function of s: and t only, and / a function of x and y only ; as, in fact, they ought 

 to be, in consequence of suppositions already made on these quantities. Thus equation (10) resolves 

 itself into the two following, 



^4^a^.l±^W4> = (It), 



df dx' ^ 



^.^H.*-/=0 (I.). 



dj^ dy' a'-' 



The equation (U) is transformable into the following: 



3-^ - --0=0 (13), 



dudv 4a. 



in which u=z+ at and v = z - at. (See Peacock's Uirampfes, p. 466). For convenience sake 

 put e for — J. Then, regarding e as a small quantity, the integral of (13) may be obtained in a 

 series as follows. 



Let — ^- = 0; then ^ = F'iti), and d> = F{u) + G («). 

 dudv du 



Hence — L^ = e \F(u) + G{v)} approximately. 

 dudv 



^= G\v) + e\F,{u) + uG{v)} 

 dv 



(p = F(?t) + G(v) + e {vFi(u) + u G, («)} ; and so on. 

 Thus (p = F{u) + Giv)+e{v F, (u) + m d (u) | + ~-^{ "" -f = (") + "' ^- i"^) 1 + &c. 



where F, (m) = fF(u) du, Fr. {u) = /F, (11) du, d (i>) = jG(v) dv, &c. Each of the functions 

 F and G separately satisfies the given equation. Let us, therefore, for the purpose of drawing 

 some inferences from this integral, suppose that F = 0. Then, 



<p = G(w) + eu G, (t)) + — -. G, («) + pY^ . G3 (u) + &c (14). 



4. It appears by this result that <p does not admit of being expressed exactly so long as the 

 form of the function G is entirely arbitrary. No inference, therefore, can be drawn from the 

 integral (14) in its general form. The nature of the series, however, suggests at once a 

 particular form of G, which gives to (p an exact expression, and which, as we shall see, applies 

 to our present enquiry ; viz. the form 4e'''. As we have already introduced the condition that 

 the velocity and condensation be small, and consequently that <p be small, whatever be x and /, 



