400 PROFESSOR KELLAND ON FRESNEL'S FORMULA FOR THE 



then the four equations in order become 



df \da.^ df 



d^J,_ ,/rf'7 (^'7/ 



<^y,_,,«.((i'y d'y,\ 

 dt' ' \dxydy:') 



Py _n^ /(^7 d^y\ ,nl_/^^d^\ pdy „dy, 

 dp 2 \da? ■*" rfy2 y + 2 V dxf ^ dyf ) d^' ' da; 



d^y,_n^ (d^y d^y\ nf UPy, d?y,\ dy dy, 

 df 2 \dx^ "^ df ) ^ 2 Vrf^; "^e^y; ; ^ dx ' dx, 



that last two equations requiring that x have the value o written for it. 



2. Since a particle at the confines of the medium must be so acted on that it 

 is in equilibriimi when 7=0, it is easy to perceive that 



'Z(f>r8x= — S(pr,Sx, taken as in equation (3). 

 This must of course arise from the variation of density near the common surface. 

 On examining the expression, it Avill appear that, when expressed in language, it 

 is equivalent to the equaUzation of the sum of a series of terms of different values, 



but of given dimensions. Now ^ dz'dx is of the same dimensions as the above 

 term ; hence we should expect that 



r 



and . . P = -P, 



r r. 



3. The solution of equation (1) is 



y=f{ax + by + ct) + ¥{—ax + hy + ct) 

 the function f coiTesponding to the incident, and F to the reflected wave ; that 

 of (2) is 



7/=/(<''/3'/+*y+<'0 



—f, (fl, x + ly + ct) 

 by ^^Titing x as the general symbol. Now we suppose the wave motion to con- 

 tinue unbroken, so that the equations (3) and (4) give the same results respec- 

 tively as (1) and (2). 



If, then, we substitute the results already obtained, we shall satisfy the two 

 equations (3) and (4). 



c"{f"[by + ct) + ¥'{by + ct)} = 



^ (a' + ^'-) {/'■ iP'J + ct) + F" {hy + ct)}+ ^V; + h-) {/;' {hy + c t)} 



+ ? {af {b y + ct)-a¥' {h y + ct)- aj; {hy + ci)} 



And the right hand side of equation (4) is the same as this. Let us now write / 

 for/(6«/ + c«) and so on, then taking notice that by (1) and (2) 



