10 Mr. O'BRIEN ON THE REFLEXION AND REFRACTION OF LIGHT 



To prove these formulas, let PP be the normal, and V ^ 



e the angle which the direction of V makes with PP', 

 then V is equivalent to Fcosfl along PP, and V sin G 

 perpendicular to PP ; therefore, since s and p are the 

 sine and cosine of the angle which PP makes with the 

 axis of !c, we have 



— = VcosO.p - VsinO.s.. 

 dt 



^= FCOS0.S+ Vsme.p. 

 dt 



• 0). 

 ..(2). 



(1) - + (2) -, and (l) 1 - (2) ? , give (since p^ + q" = 1), 



p da s dy V s da p dy V . ^ 



^ — + - -f = - cos 0, - :77 - - -77 = - - sm 0, 

 V dt V dt V V dt V dt u 



which, by the equations (A) last article, reduce to 



da dy V „ da dy V 



-— + — i- = cos 0, -i = - sin . 



dxi dai « dx dx v 



In these two equations, and in (l) and (2), put = 0, and we immediately obtain the formulae 



TT 



(B); again, put = —, and we obtain the formula (C)*- 



(3.) If M,, M2, Ms, &c....w„ be any functions of x and t, such that the equations 



Ui + U^ + U^ + w„ = (1) 



dui du^ 



dte dt 



du^ du2 



du„ du„ 



d.t? dt 



dx "' dt 



are true for all values of x and t; aj, a„, Oj, ... &c. being any -(-constants; then must 



o, = a^ = O3 = (^11 • 



^ rf(l) d(l) . , . . 



For -^-a„ -^ gives by (2) 

 ax at 



(«i - »«) -TT + («2 - a») -7- + (o„-i - ««) 



or dt 



du„_ 

 ~df 



= 



(3). 



. . d(5) d(3) . , d(2) 



d-Mi 



(a, - o„) (a, - o„.,) I^' + (as - o,,) (a^ - «„_,) ^1^ + (a„.2 - a,) (a,,.. - a,_,) 



"IF 



d'u„ 



dt' 



0. 



• The fonnuls (B) and (C) are particular cases of the following, viz. 



da da da V ^ 

 j- + T- + T- = -— cose 

 dx dy dv V 



(dfi day (dy d^y (da dy^ V^ . . . 



which may be easily proved. 



t The result of this Article is also true when a,, a,, a^ ... &c. are variables, provided they vary very slowly compared with h,, 

 ti„ »,„ &c.; in which case ^ ^ &c. will be extremely small compared with ^ ^, &c. 



