45] LAGRANGE'S DETERMINANT. 161 



changed to - i, changing any root Z to its conjugate X causes 

 every A to change to its conjugate A'. Let us accordingly write 



. I = p -f iv, A f = ^ - iv, 



A. = , + i/5 r , ^; = a r -i/3 r . 



Let us now apply the method that gave us equation 66), except 

 that we multiply the equations 64) containing I by the A"& corres- 

 ponding to A', obtaining 



69) A 2 a rs A r A' s + i x rs A r A' s + c rs A r A s = 0. 



In this equation any coefficient a rs appears in the two terms for 

 which r =p, s = t and also r = t, s =p, so that the sum is 



a pt {A p A't + A t A' p }, 

 or substituting the values of the As, 



a p t{(tt p + ifip) (at ifti) + (<x t + iftt) (ctp ifip)} 



Accordingly using a notation similar to that before employed equa- 

 tion 69) is 



70) A 2 [T() + T(/J)] + i [F(a) + JFtf)] + TT() + TT(/J) = 0. 



Now performing the same process on equations 64) with the root A' 

 and multiplying by the A'B we obtain 



71) X* [T(a) + T(/3)] + X \F(a) + F(py\ + TT() + TT(/J) = 0. 



Then A and A r are roots of the same quadratic. We therefore have 

 their sum 



so that /u, is negative. The solution therefore represents a damped 

 vibration, as in case II 43, the period and damping being the 

 same for all the q's. For another treatment of Lagrange's determinant 

 see Note V. 



Having obtained_all the ,_rQQ.ta.J L> _by substitution of any one l r 

 jjn^J^e^ji^tip^^ ratios ^A t : A 2 : - - : A n . For each 



value ofjlr we obtain a different set of ratios. We will distinguish 

 the values belonging to A r by an upper affix r, so that A r s means 

 the coefficient of e* r * in the coordinate q s . 



The theory of linear differential equations shows us that for the 

 general solution we must take the sum of the particular solutions 

 A r s e rt for all the roots A r , so that we obtain 



, Dynamics. 11 



