= 400 Correction of an oversight in Lagrange’s Formule. 
general and the most compendious method known for deter- 
mining the laws of all the small oscillations, single or co- 
existing, of every possible system of bodies, it was thought 
While on this subject, it may be as well to add, that in 
practice it will frequently be possible, an when _ possible, 
fanction of the quantities of which , J, 9, &c. are the small 
‘increments. If T remain a direct function of the three rec- 
tangular co-ordinates, it will not be sufficient to express the 
increments of these co-ordinates in linear functions of the 
small independent variables. These functions must be quad- 
ratic, or the value of V will be incomplete, and the result 
will be erroneous. But if I be transformed in the manner 
above alluded to, the co-efficients of the terms of two dimen- 
sions in the values of x, y, z, affect no part of the calculation, 
and the similar co-efficients in the integral Stim or STU Dm, 
wi g iven immediately without substitution, each of them 
by the integration of a single differential co-efficient, mul- 
tiplied by m or Dm, as the circumstances of the system may 
. happen to require. 
