e@rratt i: (ay? 22— 2a’ wo cos n’t-+-y sin n't) +a) x cosn’t-+y sin n 
ON THEORETICAL DYNAMICS. 15 
transformed equations, depends essentially upon the special form just re- 
ferred to of the function T, although, as will be seen in the sequel, there is 
a like transformation applying to the most general form of the function T. 
29. In the greater part of what has preceded, and especially in the above- 
mentioned substitutions T+U=Z and T—U=H, it is of course assumed 
that the force function U exists; when there is no force function these 
substitutions cannot be made, but the forms corresponding to the untrans- 
formed forms in T and U are as follows, viz. the Lagrangian form is 
dq_ , ddY aT 
Chi db dy ae 
and the Hamiltonian re is 
dg aT d dT 
a oP +Q; 
dt dp’ dt dq 
that is, the only difference hie that the functions Q, instead of being the dif- 
ferential coefficients with respect to the variables g... of one and the same 
force function U, are so many separate and distinct functions of the variables 
Gp +++) OF more generally of the variables g,.. p,.. of both sets. 
30. Jacobi’s letter of 1836.—This is a short note containing a mere state- 
ment of two results. The first is as follows, viz. the equations for the 
motion of a point 72 plano being taken to be 
dx dU dy _dU 
dt?” dx’ d® dy’ 
where U is a function x,y without ¢; one integral is the equation of vis 
viva 3(x'?+y")=U+hA. Assume that another integral is a=F(2, y, 2',y'), 
then 2’, 7’ will in general be functions of x, y,a,/, and considering them as 
thus expressed, it is stated that not only a'dx+y'dy will be an exact differ- 
ential, but its differential coefficients with respect to a, h will be so likewise, 
and the remaining integrals are 
r-((3 dee dy), 
ete SES, GY 
cater a), 
a theorem, the relation of which to the general subject will presently appear. 
The second result does not relate to the general subject, but I give itin a 
note for its own sake*. 
$1. Poisson's memoir of 1837.—This contains investigations suggested by 
- * Jacobi imagines a point without mass revolving round the sun and disturbed by a planet 
moving in a circular orbit, which i is taken for the plane of x,y; the coordinates of the point 
are x,y, 2, those of the planet a’ cos n't, a’ sinn’t, m’ is the mass of the planet, M the mass 
of the sun; then we ((2 accurately 
yea Gi D) }~* CGF) 
a’3 
f+ Const, 
which Jacobi suggests might be found useful in the lunar theory. The point being without 
mass, means only that it is considered as not disturbing the circular motion of the planet; 
the problem is properly a case of the problem of two centres, viz. one centre is fixed, and 
the other one revolves round it in a circle with a uniform velocity. 
