489 
poos R'N/_ sin NW cos R'N 
cosWR’ \ cosWR'sin RN J 
This being premised, then, 3°, the principle of vis viva is 
that 
m=m':m::cosRN: 
m (Rt = m' (h’t’)? ee m'(R't")? ; 
or, what is the same thing, 
RE-R"t'? sin NW cos R'N 
*¢ And 2°, the principle of equivalent moments, is that 
the moment of #’t’ round the axis AZ, is equal to the sum of 
the moments of ft and R”t” round the same axis. It only 
remains to show that these two properties are in fact con- 
tained in the Theorems 1. and 11. 
«*The pomt « is the image of W in a sphere-radius 
‘ 1 
unity. Hence, Ac = a gr — p, and, therefore, 
_ptan WR’ p*tan WR’ 
Pid Lo aa a 
D iD 
but we have, as before, sin NW=p sin RN, and consequently, 
sin? NWtan WR’ 
sin? RN —- sin? RW 
s sin? NW 
sin RWsin R"W 
‘Suppose now that the points R, R’, R", W, N, A, K, 
of Fig. 2, are all of them projected by radii through the cen- 
tre A upon a sphere, radius unity (see Fig. 3, where the 
tan WR’ =tan KV, 
tan KW= 
tan WR’. 
several points are represented by the same letters as in Fig. 2) ; 
and complete Fig. 3 by connecting the different points in 
question by ares of great circles, and by producing KW (in 
the direction from K to W) toa point Z, such that KL =90°, 
and by joining LR, LR”, and drawing the are NU’UU" at 
right angles to R”R (or, what is the same thing, with the pole 
