332 
Proceedings of Royal Society of Edinburgh. [sess. 
(a 4 sim 
(/3 2 cosv‘ 
Now, ROS is the difference between LOS and v, so that when ROS 
is minimum or maximum 8ROS = Sv ; that is, since §ROS(sec ROS) 2 
= S tan ROS 
Sv(sec ROS) 2 = S. tan ROS. 
Putting- 
2 sin v 2 + /3 2 cos v 2 - = N , 
whence 
tanLOS = ig siny j^ C ° Sy ; 
a^/P cos v + a/yJN sm v 
i -pnQ \2 _ a4 /^ 4 — 2 a 2 /3 2 e 2 . yN . sin v cos v + (a 4 sin v 2 + /3 4 cos v 2 )y 2 N 2 
(sec KQJjj) | (a 2 /3 2 + a 2 yNsinv) 2 3 
and also 
s tan K0S a 4 /? 4 + a 2 /3 2 y 2 N 2 ± a 2 j3 2 y(a 2 sin v 2 + £ 2 COSv 2 )^- 
= (a 2 /3 2 + a 2 yNsini/) 2 
so that the value of v will he determined by equating the numerators 
of these two fractions. Thus 
' 2 - a 2 /3 2 + /3 4 cosv 2 )y 2 N 2 ± 2a 2 /3 2 € 2 yN sinvcosi/ = ± a 2 /5 2 y(a 2 sini/ 2 + j3 2 cosv 2 )^ > 
' 0 V 
or 
> 9 • 9 \ 9 9 at 2 . o 202 9 at • _ a 2 /3 2 ye 2 (a 2 sin v 2 + R 2 cos v 2 ) sin vcos v 
} - a 2 sillv 2 )e 2 y 2 N 2 ± 2a 2 /3VyN sm v cos v = + ' L 
whence, after repeated simplifications, 
( - a 4 sin v 4 + y8 4 cos v 4 )y 2 + a 2 /3 2 (a 2 sin v 2 - j3 2 cos v 2 ) 
± 2a 2 /3 2 y J ^a 2 sin i/ 2 + /3 
o 9 a2 /^ 2 
2 COSl/ 2 -— Y 
cos v sm v 
a 2 /3 2 y(a 2 sin v 2 + /3 2 cos v 2 )cos v sin i/ 
J ( a 2 sin v 2 + ft 2 cosi/ 2 - — ~- 
= 0 
( - a 2 sin v 2 + /3 2 cos v 2 )y 2 | a 2 sinv 2 + /3 2 cosi/ 2 - | 2 
= + 2a 2 /3 2 y cos V sin v | a 2 sin v 2 + /3 2 cos v 2 - | 
+ a 2 /3 2 y cos v sin v { a 2 sin v 2 + /3 2 cos v 2 } 
= + a 2 /3 2 y cos v sin v | 3a 2 sin v 2 + 3/3 2 cos v 2 - 2— y- | 
