ON THE HYPOTHESIS OF UNDULATIONS. 365 



d-q I , dq''\d-q ^ dq^ d'q ^^(}_ J_^ _ p (f) 



d?~ V ~ d?) d? "^ ~ Ts' dsdt "" lis \R '^ R' j ~ ^'' 



Lastly, for q put ((> + x (0^ t''^^ function -^ (jt) being such that F (t) - ^" {f) = 0. 



rfd) dq 

 Then -^- = -r = 't a"d 

 as as 



df V ds'jds' ds dsdt ds\R R' I ^' 



If the surface normal to the directions of motion be a plane, R and R' are each infinitely great, and 

 the equation strictly applying to this case of motion, is 



'^ _ [a^.'^jEY^ + ^~—-'^^ = (4). 



dC \ ds- 1 ds" ds dsdt 



propagated in a single direction, namely, — t = F| (a + -—■) t- si; and that at the same time 



It is well known that this equation is exactly satisfied by a particular integral applying to motion 



dd) „i/ d(h\ 1 , , , . ddj 



' — J^ J I rt J- — i_ 1 / o\ • QncI fhat- at the* cnnip flmp — '— 



ds 



= a. Nap. logo. From these two equations it follows that a given state of density and velocity 

 is carried through space by the propagation and by the motion of the particles, with the velocity 



a ^ — — . The rate of propagation is therefore strictly a, whatever be the velocity and density 

 ds 



of the particles. Unless this were the case the velocity and arrangement of density in a given wave 



would change by propagation, however small the motion of the particles might be. Hence, in order 



that equation (3), in which R and R' are not supposed to be indefinitely great, may apply to motion 



in which the type of the waves remains altogether unchanged by propagation, it must be of the same 



form as equation (4). This will be the case if 



'•7Ai*h)-^'"-">'i^ <»" 



the resulting equation being the same as (4) with the difference of having a in the place of a. Also 

 2 = J. Nap. log ^. 



It is now important to remark that the general partial differential equation having \^ for its 

 principal variable, to which I have already referred, is of the third order, and consequently its 

 integral, supposing it could be obtained, would involve three arbitrary functions of the co-ordinates 

 and the time. Hence the function \^ may be made to satisfy three arbitrary conditions. The first I 

 shall suppose it to satisfy is, that the propagation of the motion be in a single direction ; the next, 

 that the motion of the particles situated in a fi.xed straight line, which I shall call the axis o{ x, he 

 entirely in that line ; the third condition I shall assume is, that for the motion along the axis of x 

 the equation (j) is satisfied. It will appear from the reasoning that follows, that a form of \|/ may 

 be found consistent with these conditions. 



Let <p^ (x, f) be the condensation at the time / at any point of the axis of « distant by x from the 

 origin, and let the condensation, for a reason that will appear afterwards, be assumed to be (f>^ (x, t) 

 X f (r, y) at any point whose co-ordinates are .r, //, x. For shortness sake I shall write (f>^ and /' for 

 these functions, treating <h^ as a function of z and t only, and /' as a function of ,f and y only. Let 

 p be the density, and v, v, w, the components of the velocity in the directions of the axes of co- 

 ordinates, at the point wyx, and at the time f. It will be assumed that u, v, and w are always 

 small velocities, and their first powers only will be taken account of. Tliis being ])remised, we have 



