12 Mk. O'BRIEN ON THE REFLEXION AND REFRACTION OF LIGHT 



SECTION II. 



The general Equations of Con7iection of the Vibratory Motion of two Elastic 

 3Iedia, separated hy a Plane Surface. 

 (7 ) The two media are supposed to consist of discrete particles symmetrically arranged, 

 and acting upon each other with forces varying according to any law which ensures stable 

 equilibrium By the Surface of Separation, we simply mean an imaginary plane described 

 between the two media, the particles of one medium lying on one side of it, and those of 

 the other on the other side. In the immediate vicinity of this plane, the media are supposed 

 to exercise a mutual repulsion, so that no mixture takes place. We shall take the plane of 



separation to be the plane oi xy. . „ , ., • ,. . 



(8.) We shall obtain the general Equations of Connection of the vibratory motion of the 

 two media, by means of the following self-evident Principle. 



When a very small vibratory motion is communicated to a stable system of particles, such 

 as the two media just described, we may assume that the vibratory motion will always remain 

 very small, and, at most, of the same order of magnitude as the original motion. 



This principle is either tacitly assumed, or employed as self evident, by all the writers who 

 have treated of the problem of the transmission of waves from one medium into another. Poisson 

 states it very clearly in the Mimoires de I'Institut, Tom. x. p. 320, and makes use of it precisely 

 as we shall do in the present paper. It is evidently assumed in the Article Sound, Ency. Metrop. 

 p. 776; for by saying that the two media must have a common elasticity at their junction, and 

 that that elasticity is expressed by £ (l + /3«), and E\l + ft's'), the writer supposes that there 

 is the same slow variation of elasticity at the surface of junction as elsewhere, and therefore the 

 same slow variation of pressure, and consequently the same small vibratory motion. 



(9.) To apply this principle to the case we are at present concerned with, let wyz (» = 0) 

 be the co-ordinates of the equilibrium position of any particle (P) of the lower medium in the 

 immediate vicinity of the plane of separation, a, /3, 7 its displacements at the time t, and let 

 .r; +&X, y + Sy, z + Sx, a + ^a, /3 + 5/3, 7 + S7 be the co-ordinates and displacements of any 

 other neighbouring particle (Q) of the lower medium ; also let w + Ax, y + Ay, x: + A«, a + Aa, 

 /3 + A/3, 7 + A7 be those of any particle (F) of the upper medium. 



Put r^=lv'+Sy'+Sz', and r'"^ Ax^+ Af+ Ax\ 

 and let w/(r), m'(p{r') be, respectively, the forces exercised by Q and P on P. Then, if ^ be 

 the whole force, parallel to the axis of x, brought into action upon P by the vibration, we have 

 (see Cambridge Transactions, Vol. vii. p. 403) 



X = -2m {fir) Sa + ^-f (r) Sw {SxSa + SyS(i + SzSy) } 



+ 2'»»' {0(r') Aa+ -,(})' (r) Aw (Ax Aa+ AyA(3 + A«A7)|, 



2 referring to the lower medium and 2' to the upper. 



In this expression we shall substitute for 5a the series 



da :, da t, rfa » 



— Sx + --hy + -r-ox + &c 



dx dy dx 



Also, let a, /3', 7' be the values which the displacements a -I- Aa, /3 + A/3, 7 + A7, assume 

 when X, y, « (=0) are substituted in them ior x + Ax, y + Ay, z + An, then we have 



a + Aa 



da . da . da . 



■■ a + -r- Ax +-— Ay + -r-Az + &c. 



n .1! // « n.« 



Now the differences of the corresponding displacements of two contiguous particles at a distance 

 from the plane of separation must be indefinitely small, (supposing of course, as is always done, 



