16 Proceedings of the Royal Irish Academy, 



Therefore we have, finally, 



( mxRX"^ I mxQ\^ 



which we may write, for convenience, in the form 



^V==(^.+ (^, (13) 



where a and /3 are positive, since b', c', and A are positive. 



It can be shown as follows, that this equation cannot have more 

 than one negative root. 

 Assume a > /?, and let 



k' = k- 0; 

 then we have 



mV-^'^ (^' - yV = <f^<:"^ + »'^ (^' - 7)^ 



where y = a - )8, and is therefore positive. 

 Expanding, we have 



juV^'* - 2ix^p'^yk'^ + (;u2j»272 -q^- r^) ^-'2 + 2r27^-' - r^^^ = 0, 



which equation cannot have more than one negative root, by 

 Descartes' Eule of Signs, since y is positive, and, a fortiori, equa- 

 tion (13) cannot have more than one negative root. 



If jji is the coefficient of friction, which is just necessary to prevent 

 motion at the point, the last term of equation (13) is (^* - ii!"^) p'^o?^'^, 

 and therefore if /a > fc', the equation has no negative root. 



Geneealized Co-oedin^ates. 



"When there are several bodies in the system, it will often be 

 more convenient to use Lagrange's generalized co-ordinates. Suppose 

 the system has n degrees of freedom, and that there are m smooth 

 constraints. It would then have m + w degrees of freedom if the 

 constraints were removed. Let 6i, 0^ . . . On, ^i, ^2 • • • <^m be the co-or- 

 dinates of the system when the constraints are absent, ^i, ^2 . . . <j!),„ 

 being the distances of the constrained points from points fixed in 

 space, so chosen that, in the position of rest, the lines joining the 

 fixed points to the constrained points coincide with the direction of no 

 motion. The ^'s may be called the co-ordinates of constraint. "We 

 will subject the co-ordinates to the condition that the rectangular 



