DIFFERENTIAL EQUATIONS OF DYNAMICS, ETC. 
3\9 
dP 
tion required. For we have, at once, ; also 
dP d{x■^) d{x^) d{xn) . 
+ ■ • • ’ 
now the definition of rii is where 
W= — (Z) +x;?/ 2 + . . . 
(expressed in terms of li, li, ...11 (art. 66 .)) ; hence, putting Z (without brackets) 
for the original form of Z in terms of &LC.,y^, &c,, and observing that (Z) becomes 
a function of &c., only through y^, &c., we have 
dW dTi dy^ dTi dy^ d^L dyn 
d^~~dy^k’~~ dy^ d^. ~ dyn d^'. 
j^x—4-x^A- - 1 -jr' — 
dx. , dx„ , , dx' 
+y'ds\^y‘w^^-'^y'‘W' 
dZ 
d{xj) , d{x{)^, , d{x,)^, , , d{xj)^, 
= -TF + -sr 5. + -ir 5* + • • • + 
dx\ d[x,) ^ 
d^n 
,. ~ . . . . . , aij . 
The two first lines of this expression vanish by virtue of the equations Xi=-^ ; and 
dt 
we have 
so that we have, finally, 
d\\ 
u,,, d{xA , d{xo) , , dixn) 
which is evidently equivalent (see the expression (82.)) to 
dP . 
''‘^~dh' 
the proposition in question is thus established, and may be enunciated as fol- 
lows : — 
70 . Theorem IX . — Every transformation of coordinates is a normal transformation, 
of which the modulus * V is a function of |i, ... 3 / 1 , {with or without t) given hy 
the equation p_ {x^jy, + {x^)y ^-\- . . , -f {Xn)yn 
{the brackets indicating Xj, ... x„ ore to he expressed in terms 0 / ?,,... |„). (See 
arts. 62, 68 .) 
This theorem will be made more intelligible by applying it to a very simple 
example. 
Let it be proposed, then, to transform from rectangular to polar coordinates the 
differential equations of any dynamical problem referring to the motion of a single 
material point whose mass is m. Let x, y, z be the rectangular coordinates of m. 
* See note on Theorem VIII. art. G4. 
