358 Mr KELLAND, ON THE TRANSMISSION OF LIGHT 



Then sin^m' = 1 — eos^m' 



= 1 — (cos^'coswi + sin<p' sinjncosOf 

 sm"n' = 1 — (cos0'cos?« — sin (p' sin m cos 9)" ; 

 .-. sin^jw' sin-?;' = (1 — cos^^'cos^;« — sin=0' sin^;» sin-^f 



— 4sin*^'cos^<|)'sin^»7«cos-»KCOs^0 

 = \sin''<p' + siii'm - sin'^'sin^w; (1 + sin'^fl)}' 



— 4 sin^(/)' cos-</)' sin^w^ cos^?n cos^O ; 

 sin'asin^/3 = {sin'(p + sin' Ts - sin-^sin^Ta: (1 + sin=0)p 



— 4 sin°^ cos'^ sin- Tz cos- T'z cos'-' 6* 

 = // j[sin^^' + sin^w - M"sin-<^'sin-«/ (1 + sin-0)]" 



— 4 sin^ ^' sin- »« cos' (/) cos" 7'ss cos^ 6 ] . 



Now if <l>' and m be both small, this expression becomes 

 sin^asin*/3 = M'(sin'>« + sin'0' - 2sin^>«sin'0' cosSei), 

 and sin^»?'sin^«' = (sin' >« + sin' (^' - Ssin'wisin^^'cosaf^); 

 .". sin'asin^/S = M*sin'»«'sin^H'. 



If m be very small compared with 0' 



sin" a sin" /3 = lu'sin'cj)' 



= ;tt'sin"»e'sin-«'. 



If <{>' be very small compared with ?«, 



sin"asin''/3 = /u'sin'w = ^^sin-w'sin^w'. 



In all cases therefore, provided one of two, either cp' or i/i be small, 

 or if they are both small, we have 



sin a sin ^ = fx' sin m' sin n' ; 



