G64 
PROFESSORS A. W. REINOLD AND A. W. RUCKER 
Again, since 
Y 2 —/3, 2 sin 2 4]~h a i 3 cos 2 </>„ 
d/3, sin 2 </>,-\-da, cos 2 <^=0 'i 
and similarly >.(22) 
d/3. 2 sin 2 (f) 2 J r da 2 cos 2 <f> 2 =0 J 
Equating the two expressions for the square of the common ordinate at II, we get 
d/3, sin 2 xp,-\-da, cos 2 xjj,=:d/3 2 sin 2 i \f 2 J r da 2 cos 2 i/q.(23) 
Since, by Beer’s equation, 
dy __ \/ (A-?/)(?/ -f3~) 
dx y 2 + a/3 
we have, by equating the expressions obtained by substituting the values of y in terms 
of cq, /3„ and xjj,, and a 2 . (3 2 , and \fj. 2 , 
(da,— d/3,) sin 2 xjj, = (da. 2 —d/3 2 ) sin 2 xfj. 2 . .(24) 
Substituting from (20) in (23) and (24), 
d/3, sin 2 (<p,-\-g,)-\-da, cos 2 ((/),+ g,)=d/3 z sin 2 (<£0 — Q+da 2 cos 2 (</>o—£,)> 
and 
(da,—d/3,) sin 2(<f>, + £,) = (da 2 —d/3 z ) sin 2(c/) 2 — g,). 
Again, substituting in these equations from (22), 
and also 
therefore 
da., sin 2 (/>0 cos 2 (cp. 2 — | 2 ) — cos 2 <£ 3 sin 2 (<£ 3 — | 2 ) 
da 2 sin 2 cf), cos 2 (</>, + |d - cos 2 (p j sin 2 (<p, + |d 
X 
sin 2 cf), 
• Of? • 
Sill'- 90 
sin 2(0 2 —1 3 ) sin 2 </ >, 
sin 2(</q + Id sin 2 $3 
sin (2</> 3 —1 3 ) sin | 2 _sin 2(cft 2 —1 2 ) 
-sin (2<fo + |d sin sin 2(0! + ^)' 
(25) 
(2G) 
Also from (21) and (24) 
dV/TrY' 2 =2g 1 (da,-{-d{3 1 )-{-2£ 2 (da 2 -\-d/3 2 )— (da, —d/3,) sin 2(j),-\~ (da 2 —d/3 2 ) sin 2<£ 3 
= sih?^( sin 2< ^” 2 & cos 2(/> z ) - gi J^(sin 2</>, + 2g 1 cos 2&).(27) 
