containing 4, 5, 6, 7, and 8 corpuscles, etc. 43 



Case of Five Corpuscles. 



One position of equilibrium is when four of the corpuscles are 

 at the corners of a regular tetrahedron while the fifth is at the 

 centre of the tetrahedron. 



Let x, y, z be the coordinates of the centre of gravity of the 

 five corpuscles, f lf r) 1} £, £,, rj 2 , £>, £ 3 , r} s , £ 3; £,, ?? 4 , £, the coordi- 

 nates of the four outer corpuscles relative to the central corpuscle, 

 the direction of the axes being the same as in the last example. 



Then if 



£i + & + & + I4 =p, Vi + V* + Vi + Vi = q, £1 + £2 + £3 + t* = r > 

 & + &-(&+f«)=.Pii »h + »7a- (^1 + 174) = ?i, Si +&-(£■+ f4>=n, 



^i + ^i-(^+.^s)=Ps, Vi + Vi-(V2 + V 3 ) = qi, £+&-(&+ ft) = r„ 



we have 



d 2 x , d' 2 y 7 d 2 z , 



m -=- = — few, m -^z = — ky, m -7— = — kz. 

 dt 2 dt 2 u dt 1 



Thus, x, y, z are principal coordinates, the time of vibration 

 being 2*^%. 



We have also 

 d 2 

 m dt 2 ( pi + < i 2+r ^ = ~ kl (P. 1 + V 2 + r ^' 



d 2 d 2 



™-£p(Pi-q*) = -h(Pi-q2) and m^(p 1 -r 3 ) = -k 2 (p 1 -r 3 ), 



3 



where *, = 3*, k = -J^-k. 



3V3 + 4V2 



Thus pi + q 2 + r 3 , p 1 — q 2 , p l — r 3 are principal coordinates. 



Again 



d 2 d 2 d 2 



m-^ 2 (p2-qi)=0, mjp(p t -r 1 ) = Q, m ^ 2 (q 3 - r 2 ) = 0. 



Thus p 2 — q l} p 3 — r x , q 3 — r 2 are also principal coordinates. 

 The remaining six principal coordinates are 



r + \(p 2 + qi), r + Xoip.z + qJ, 



q+-\ in + Ps), q + ** ( r i + Pi\ 

 P + \ (q s + n>)> P + x 2 (ft + r 2 ), 



