504 PROFESSOR KELLAND ON THE THEORY OF WAVES. 



motion. To solve the problem more generally, we should assume for z a series 

 of sines of multiples of 6. If we restrict ourselves to two sines, we shall have 

 the following equation : 



2"^ 

 z = h + asm. -r— {ct—x) + esai.2 6 ; 



which gives 



1/ dz y 2'7r 

 z' dt « " A 



(a cos 6. c — ; — h 2 e cos 2 5 . c —\ 

 dt dt) 



du 2'ir ( n (Ix -, n a '^^\ 



-— = — :r — ( a COS V . c — — •+ zecosz v .c — — 1 . 

 dz \z \ dt dt) 



The same limitation as to the extent of the series gives 



u = b sin 6 +f sin 2 0. 

 Diiferentiating this, and equating it to the former, 



6cos0 + 2/cos20 = i(acos0+ 2ecos20) (c-~\ ; 



(6 cos + 2/ cos 2 0) {h + asm6 + esm26) 

 = {a coaO + 2e CO&26) (e — 6 sin 0—/ sin 20), 



or bh COS0 + 2fh cos 2 + ^ sin2 a + a/(sin 3 0-sin 0) + ^ (sin3 6 + s\n6) + e/sin4 

 = accost + 2cccos20 — -— sin 2 — e 6 (sin 30 — sin 0) — -j^(sin3 + sin0) — e/sin40. 



By equating the coefficients of cos 6 and cos 2 6, we get 



bh-ac,2fh = 2ec; 

 hence we learn that, if z be known, u is kno'svn. 



With respect to the terms involving sines, it is clear that no equations can 

 be made, since the same quantities will occur again. 



6. We now proceed to determine the motion pai-allel to the axis of x by an 

 independent method. We will conceive the portion PQ to become solid for an 

 instant, and calculate the force by which it is urged in the direction of the axis 

 of X. That force wUl consist of two parts, totally independent of each other ; the 

 one the difference of statical pressure on the two planes PM, QN ; the other the 

 difference of the dynamical impulses on the same planes. 



With respect to the latter, it may be remarked, that it must be absolutely 

 independent of v. The real effective pai-t of this pressure is, in fact, nothing more 

 than the resistance on either of the planes due to the velocity of the fluid in a 

 direction at right angles to it. 



In order to obtain the value of this force, we shall not attempt to treat of it 

 separately as a problem of resistances, but apply directly to it the same argimient 

 as is commonly applied to establish the theory of resistances. We have akeady 

 stated that it must be independent of c : this can only be on the supposition that the 



