DIFFERENTIAL EQUATIONS OF DYNAMICS, ETC. 
t3.55 
If Xj, ... x^ be the coordinates employed in the first statement of any dynamical 
problem, the differential equations are comprehended in the formula 
^wy 
dx'.) 
dxi 
(D.) 
[If there be any forces, such as those arising- from a resisting medium, which do 
not satisfy the natural conditions of integrability, then on the right-hand side of the 
formula (D.), instead of 0 we shall have an expression such as ^i(K^Xi) ; but such 
terms are easily introduced and allowed for separately, and do not affect the follow- 
ing investigation. I shall therefore here assume that they do not exist.] 
In the above formula, W is a given function of x^, ... x^, j?', ... which may also 
explicitly contain t. 
In Section VI. the only case contemplated was that in which ... x,, are inde- 
pendent coordinates ; in which case tlie formula (D.) is equivalent to n separate equa- 
tions, since Bj?!, &c. are wholly arbitrary and independent. 
In practice, however, it is often more convenient to use, at first, a set of coordinates 
more in number than the independent variables of the problem, and therefore subject 
to equations of condition. 
Let us assume then that the n coordinates x^, ... are subject to r equations of 
condition, 
Lj=0, L 2 — 0, ... L^=0, 
where Lj, &c. may explicitly contain t, besides the n variables a;,, &c. 
W 
If we introduce the n conjugate variables 3/,, ...y^ defined by the equations 
•Aj i 
and take Z a function of Xi, &c., 3/1, &c. (with or without t), defined by the equation 
Z= — [W] - 1 - [x\']y, -H . . . + [x'^yn 
(the brackets indicating that/j, &c. are expressed in terms of 3/1, &c.), then it follows 
exactly as in art. 18 (Part I.), that the formula (D.) will be changed into the system 
dZ 
[In the most usual problems W is of the form T-j-U, where T is homogeneous and 
of the second degree in x\, &c., and U does not contain x\, &c. at all. In this case 
Z is only T— U expressed in terms of y,, &c., instead of x[, &c. But T is not neces- 
sarily homogeneous ; in fact it is not so in problems relating to motion relative to 
the earth, as affected by the earth’s rotation,] 
Let us now suppose that the system (E.) is to be transformed by the introduction 
of the m independent coordinates ... and of the new conjugate variables 
’Ju ^2j ••• ; where m~n — r. And let it be required to investigate a theorem by 
means of which the transformation may be effected without recurring to the original 
formula (D.). 
MDCCCLV. 3 B 
