Correction of an oversight in Lagrange’s Formule. 399 
dinates their values in the state of equilibrium, substituti y 
tions of ail 
or their increments equivalent quadratic func 
independent variables, and then integrating with respect to 
the dimensions of the system. In doing this, it is evident, 
that in order that nothing may be neglected which may af- 
fect the terms of two dimensions of £, L, 9, &c. in the value 
of V, (the only terms in that function which it is necessary to 
attend to,) the terms. of the second dimension which enter in- 
to the value of the increments of the co-ordinates, must be 
included in the pe wr by all the frst differential co- 
efficients in the developement of F, although they may be 
disregarded in the sribgeititai for the prec and proguets 
of the increments of the co-ordinates. will then 
seen; that to render the co-efficients [1], [2], &e. pe oa 
it will be necessary to add to them the following i int tegrals: __ 
To [1] (Gert ah +a )m 
(2] as (FZ ag 4 42 pl2+ i es)m 
Br. sq “2% a mUs+ Gees jm 
&ce. 
To [1,2] : “s (nes ina a2) 
[1,3] s (Ga3+% ats oe 
[2,3] s(G25+5 Fog 4 o 023) 
where a'1, a’2, a/3, &c. a1,2, al,3, 02,3, &c. are the co-effi- 
cients res ctively of 2, 1, p?, &c. EL. £9, Lo &e. in re 
lee a f x; 0/1, &c. of 37, &c. — rye = 
& 2 Bee! in the general value of z. (p- 3 
2 he ot which is half the sum of the living forces 
of the system, is evidently, in the case of small osci > 
ted by the above omission. 
eo  eieaeiale’ in the foregoing formulz, are, no ss 
such as every practised mathematician who has c a = 
this chapter of the Mécanique Analytique, has alrea “i mee “ 
for himself ; but as the error is, at all events, to be sdemon — 
the last edition of the first volume, that of 1811, — 
formule in question aflord by far the simplest, 
