489 
yop RRA pcos R'N R'N _ sin NW cos R'N 
fasnic cos WR’ \ cosWR' sin RN] 
This being premised, then, 3°, the principle of vis viva is 
that 
m (Rt)? = m'(R't’)? + m'"(R't")? ; 
or, what is the same thing, 
R?-R't” hi m' éf sin NW cos R’N 
Rt?  m_ cosWR'sinRNcos RN 
«« And 2°, the principle of equivalent moments, is that 
the moment of R’t' round the axis AZ, is equal to the sum of 
the moments of Rt and Rt” 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 point « is the image of W in a sphere-radius 
s 1 1 
unity. Hence, Ac = > kW a — p, and, therefore, 
ptan WR’ _p* tan WR’ 
1 1-p 
ie 
but we have, as before, sin NW=p sin RN, and consequently, 
sin? NW tan WR’ 
sin? RN — sin? RW 
t sin? NW 
sin RWsin R"W 
‘¢ Suppose now that the points R, R’, R”, W, N, H, K, 
of Fig. 2, are all of them projected by radii through the cen- 
tan WR’ = =tan KV, 
tan KW= 
tan WR’. 
tre A upon a sphere, radius unity (see Fig. 3, where the 
several points are represented by the same letters asin Fig. 2) ; 
and complete Fig. 3 by connecting the different points in 
‘question by arcs 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 RR (or, what isthe same thing, with the pole 
