490 
H) meeting LR', LR, and LR” produced, in the points U’, 
U, U” respectively. By what has preceded, the points K, L 
of Fig. 3 are constructed 
precisely in the same man- 
ner as the same points in 
Fig. 1, and in fact Fig. 3 
is nothing else than Fig. 1 
with some additional lines 
and points. The condi- 
tion employed to deter- 
a 
Uv 
mine the magnitude of the 
vibrations Rt, R't’, R’t’, 
gives that these vibrations 
are as Fig. 3. 
Sun) rainy, pele ets 
or, observing that LA, LR', LR” are the great circles whose 
poles are 7, 7”, T” respectively, these vibrations are as 
sin R"DLR': sin RLR": sn RLR’; 
and, substituting these values, the equation given by the prin- 
ciple of vis viva becomes identical with that of Theorem 1. 
‘* Proceeding to the condition given by the principle of 
equivalent moments, we have 
moment of Rt round AH 
=Rtx ARxcos | AR, | dist (Rt, AH)] x sin (Rt, AH); 
and in Fig. 3, observing that the radius through U is parallel 
to the perpendicular distance of (Rt, AH) (for LR has the 
pole 7, and NU the pole H) then 
cos [AR, 1 dist (Rt, AH)]=cos RU, 
sin (Rt, AH) =sin TH, 
or, since T and Z are the poles of LR and NW respectively, 
TH = 4 NUR, and, putting AR = 1, the moment is 
= Rtcos RUsn NUR. 
Similarly, 
