28 Mr. O'BRIEN ON THE POSSIBILITY OF 



must investigate the equations of connection of the vibratory motion of two media separated by a 

 plane (as in a previous paper in the present part of the Cambridge Transactions), supposing that 

 one of the media is composed of material as well as ethereal particles. 



We shall make the same suppositions, and use tlie same notation, as in the paper just referred 

 to; assuming the upper medium to contain material particles, and a, /3,, 7, to belong to any one of 

 them, a, (i, 7, and a, (i\ 7' belonging (as before) to the etiierial particles. 



Then tlie force parallel to the axis of x on any particle of ether at the plane of separation 

 will be 



da „ rf'y ,„, ^'^ da ^, dy . . 



(C + D) — + D — - - (C + D ) D ■— + terms of superior order. 



dx dx dx dw 



The terms of superior order here alluded to consist, in the first place, of higher differential 

 coefBcients of «, (i, 7, a, /3', 7', and secondly of terms arising from the action of the material parti- 

 cles, the largest of which we have assumed, in obtaining the equation (l), to beef the same order 

 of magnitude as the second differential coefficients of a, /3, 7. Hence we have at the plane of 

 separation 



(C+i))^ + Z>^=(C'+iJ')^ + Z>'^' (2). 



dx dw dx dw 



In the same way we obtain 



iC^D)f^D'^ = (C'^D')'-f^D'^ (3). 



dx dy dx dx 



^C^E)'-l^D(p^f)=iC'^E')'-L^n'(^^f) (4). 



dx \dx dzj dx \dw dy / 



We also find, just as before, 



a' = a, /3'=/3, 7=7 (5)- 



In addition to these six equations of connection, we obtain three others in the following 

 manner. 



At the plane of separation the force acting on any particle of matter is 



2m {^(r) Sa, + - ^{,'(r,) Sx, (Sx^ Sa, + Sy, ^[i^ + Sxjy) } 



+ a part arising from the action of the ethereal particles. 



This may be reduced, as the force on an ethereal particle, to the form 



da dy 



(C, + D') -~ + D, -^ + terms of superior order ; 

 dx ' dx 



observing that we include the part arising from the action of the ethereal particles among 

 the terms of superior order for the same reason as before. We have, therefore, at the surface 

 of separation, 



dx dx 



and similarly (C + J> ) — ' + D -^ = 



' ' dx ' dy 



' dx ' \dx dy J 



.(6). 



dy . 

 These nine equations, namely, (2), (3), (4), (5), and (6), are the complete equations of 



