10 REPORT 1869. 



Sketch of a Proof of LKjranjes Equrdion of Motion referred to Generalized 

 Coordinates. By II. B. Haywakd, M.A. 



Suppose a material system with n degrees of freedom, i. e. one for the determi- 

 natiou of whose position and configuration ;; absolutely independent variables or 

 coordinates are necessary and sufficient. Denote one of these coordinates b}' q, and 

 suppose q to be changed to q-\-^q ; then a given particle (mass m) of the system re- 

 ceives a displacement definite in intensity and direction, which may be denoted by 

 kbq, where k may be regarded as a magnitude definite in intensity and direction, 

 which may be called the " variation coefficient " of the particle m with respect to 

 the variable q. It is plain that /■; is in general a function of all the n coordinates. 

 If V denote the velocitj^ of the particle «i in any possible state of motion of the 

 system, T(or gSwy-) the vis liva of the system, and q the rate of change of q or the 

 diffiirential coefficient of q with respect to the time, the author proves that, suppo- 

 sing T expressed in terms of the n coordinates q &c., and their differential coefficients 



with respect to the time q' &c., and -— denoting the partial differential coefficient 



of T with respect to q, 



-— ='S,(jnhv cos ^), 



where (/) is the angle between the directions of h and v, and the summation extends 

 to all the particles of the system. The quantity 2(7nJ;v cos<^) may be appropriately 

 termed the jxirtial momentum of the system v\-ith respect to the coordinate q ; and 

 it is easil)' seen that the ii partial momenta with respect to the n coordinates com- 

 pletely determine the motion of the system. It thus appears that each of the well- 

 known equations of motion in the Lagrangean form 



./.^ 



rlq' _(1T _cn5 



dt dq dq 



does but express the relation between the rate of change of one of the partial mo- 

 menta with the time and the forces acting on the system. 



This interpretation immediately suggests a direct proof of Lagrange's equations 

 without any reference of the system to fixed rectangular or any other special system 

 of coordinates as in the proofs hitherto given. The direct result of diti'erentiating 

 2(wiAycos (^) with respect to the time t is reduced to the required form by the help 

 of the equation 



dv dk / 7 • / 

 -r- = -3- cos <h—k sm d) . «, 

 dq dt ^ ^ ' 



where et is the rate'of change of the direction of A; in the plane of v and k ; a relation 

 which it is not difficult to establish. 



On Curves of the Third Der/ree, here called Tertians. Bi/ F. W. Xewmax. 



The object of this paper was chiefly to suggest a nomenclature for those curves 

 of the third degree which are diametral or centric ; but to make the argument 

 clear, a concise discussion of the curves was necessary, and a paper was annexed on 

 the roots of a cubic equation. 



If r/a;'' + 3/3a:- + 37X+S=0 be any cubic equation, and 



A = 



a/3 



B= 





2D= 



a/3 

 yfi 



it is here shown that if AB — D-.> and also A < 0, there are three real unequal 

 roots to the equations. But if A<0 and A — BD-=0, there are two equal roots. 

 Lastly, if either A-'O orB>0, or AB — D-<0, there is only one real root. Suppo- 

 sing T|-|-Tj+T, + C = to be the general equation, where 



T, = ccx' -)- ^^x'y + 3yxf^ + Bf, 



