484 
cos NR’- cos NWeos WR’ | 
ae sin NW sin WR ; 
and therefore, 
2 sin? NW tan iE oe 
pave gg 2 e 
cos Wh C8 NF sin 2 NW 
or the expression becomes, 
ie. 9 sin NW 
sin R’R ~ cosWR' 
And sin R”R =sin2 NR =2sin NR cos NR, so that finally 
the expression becomes 
cos NR’. 
ie sin NW cos NR’ 
~ cos WR’ sin NR cos NR’ 
which proves the theorem. 
‘* To prove the second theorem, take as before R, R’, W, 
to denote the angles LRR’, LR"R, NWR’, respectively ; 
and moreover, U, U’, U” to denote the angles NUR, NU'R’, 
NU"R", respectively ; then considering, first, the function 
on the left-hand side, viz. : 
sn R”LR' cos RU sin U+sin RLR' cos R’ U" sin U", 
we have 
sin NR 
sin RU’ 
cos RU sin U=sin NR cot RU 
= sin VR cos R cot NR =cos R cos NR, 
and in like manner, 
sin U= 
sin VNR” 
sin R”U” 
cos #"U" sin U" = sin NR” cot R"U" 
=sin NZ” cos R"cot NR” = cos R” cos NR” = cos R"” cos NR, 
and the expression thus becomes 
=cos NE (sin R” LR’ cos R + sin RLR’ cos RK"), 
which is 
sin U" = 
