328 
Proceedings of the Royal Society of Edinburgh. [Sess. 
by an examination of the logic of the process of deduction, that the varied 
motion was not in all cases what it was tacitly assumed to be, in itself a 
possible motion — that is, one consistent with the kinematical conditions of 
the system. What we are entitled to assume is only that if the motion be 
slightly varied from a configuration C in the actual succession of configura- 
tions to a corresponding configuration C' in a neighbouring succession, 
the variation from C to C' must be consistent with the conditions of the 
system. Holder examined the cases for which, as Hertz pointed out, 
the method of least action seemed to give erroneous results, and showed how, 
by a stricter logical process than that usually employed, they could be 
brought likewise under the scope of the principle. 
4. To make clear how the method fails, I shall consider the process of 
derivation of Lagrange’s equations from the equations of motion of a free 
particle — a process first employed, so far as I am aware, by Lord Kelvin. 
Exact equations of the kind first indicated by Ferrers, applicable to all 
cases of motion in which the co-ordinates are such as to explicitly define 
the configuration at any instant, will be obtained ; and then I shall inquire 
how the method of ignoration of co-ordinates, and the gy rostatic equations, 
first given by Lord Kelvin, are to be modified when this is done. This 
inquiry is the principal object of the present paper, and it will be seen leads 
to a simple result. Solutions will be added of a few problems illustrative 
of the methods arrived at, and of the sources of errors that may easily arise 
in their application. 
5. The relations between the x, y , 0 co-ordinates of a representative 
particle and the independent parameters q v q 2 , ... , q k may be written in 
the form 
Sx = a l Sq l -f a 2 Sq 2 + ....+ a k Sq k \ 
+ •••■ + hh* • • ( i ) 
Sz = c^q x + c 2 Sq 2 + .... + c k 8q k ‘ 
for a displacement possible at time t. 
In the case of finite equations of condition the coefficients a v a 2 , ... , 
b v b 2 , ... , c v c 2 , . . . . are partial differential coefficients of functions of 
the co-ordinates q v q 2 , . . . , q k : in every case they are functions of the 
co-ordinates. The real displacements for an interval of time dt are 
dx = a l dq 1 + a 2 dq 2 + ....+ a k dq k + adt j 
dy — b 1 dq 1 + b 2 dq 2 + . . . . + b k dq k + bdt . . . (2), 
dz = c l dq l + c 2 dq 2 + ....+ c k dq k + cdt " 
where a, b, c are zero if the constraints do not depend on the time. It is 
supposed that the coefficients are so chosen that equations of condition ex- 
