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- 
mine the magnitude of the 
vibrations Rt, R’t, Rt’, 
gives that these vibrations 
are as 
aM 2 sein.) sd 
or, observing that LR, LR’, LR" are the great circles whose 
poles are 7, 7”, T” respectively, these vibrations are as 
sin R”"LR': sin REAR”: 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, 1 dist (Rt, AH)] xsin (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, | dist (Rt, AH)]=cos RU, 
sin (Rt, AH) =sin TH, 
or, since 7’ and H are the poles of LR and NW respectively, 
TH = 2 NUR, and, putting AR = 1, the moment is 
= Rtcos RUsn NUR. 
Similarly, 
