40 Prof. Thomson, On the vibrations of atoms 



where 



P = & + £ + £ + f4 3 g = % + ^2 + */ 3 + 1/4, *" = £l + £ 2 + £j + &, 



and ^ = "&3~ ; 



thus p, ^, r which are proportional to the coordinates of the centre 

 of gravity of the four corpuscles are principal coordinates and their 



vibrations give three periods each equal to lir a / x • 

 If 



2>i=£i + &- (& + &). qi = vi + V2-(v3 + Vi)> r 1 = Si + C a -(t, + p, 



i>2=|l + ^3-(^2 + ^) ) g'2 = 1/l + 1/3-('/2 + '/4), ^ 2 = & + £ 3 - (& + £ 4 ), 



B» =£ + £4 -(& + &), g r s = % + ^4-(^a + %) J »»=£ + &-(£, + £,), 



we find 



e£ 2 eZ 2 d 2 



d 2 d 2 



m dtAP2 + <2i) = - k (p* + q0> m ^ (p» + n) = - *i ( j5 3 + n), 



rf 2 d 2 



m dt 2 ^ 3 + r ^ = ~ kl ($ 3 + r ^> m dt 2 ( pi + ^ 2 + r ^ = ~ ^ (Pi + < 2 2 + r *>> 



d 2 d 2 



m ^ 2 (Pi-^) = -h(pi- q*)> ™> op (Pi -r») = -k t (p l - n), 



6e 2 

 where ^i = T? = l^> 



12e 2 



k 2 = -p- = 3&, 



S^ 2 



K 3 — ^ 3 — 4 tf. 



Thus the remaining nine principal coordinates of the system, 

 which has twelve degrees of freedom, are 



Pi — <Zi> p» — ^i, q.z — 1'z, the three corresponding periods of 

 vibration are infinite ; 



Pi + Qi, Ps + r 1) q s + r 2 , the three corresponding periods of 



vibration are 2tt a / -j- - ; 



