SN 6 re ae ee 
ON THEORETICAL DYNAMICS. 21 
formule. But Jacobi proceeds (and this is given as entirely new) that the 
disturbed equations being 
@x dv do niv= dU da dz dU dQ 
"dE de! dx’ de dy ' dy? "dB dz | de’ 
then the equations i the variations of the above system of arbitrary con- 
stants are 
dps dQ dr dQ. 
dt~ da’ dt dh’ 
so that the constants form (I think the term is here first introduced) a 
canonical system. 
Jacobi observes, that in the theory of elliptic motion, certain elements 
which he mentions, form a system of canonical elements, and he remarks, 
that since one complete solution of a partial differential equation gives all 
the others, the theorem leads to the solution of another interesting problem, 
viz. “ Given one system of canonical elements, to find all the other systems.” 
This is effected by means of the second theorem, which is as follows :— 
II. Given the systems of differential equations between the variables 
a(a,,a,..a,,) and b(d,, b,...6,,) 
da dH db_du 
db peee dt da’ ee 
where H is any function of the variables a,... and b,...; and let 
a(a,,a,...a,,) and 6((,, 3,..(,,) be two new systems of variables connected 
with the preceding ones by the equations 
where wy is a function of a,...5,... without ¢ or the other variables, then 
expressing H as a function of ¢ and the new variables a,... and f,. 
these last variables are connected together by equations of the like form 
with the original system, viz. :— 
de__dil  dp__ at 
ad dp’*** GE. tantadae 2. 
Jacobi concludes with the remark, that other theorems no less general may 
be deduced by putting Y+Avy,+py.+ ... instead of J, and eliminating the 
multipliers A, »,.. by means of the equations y,=0 ,,=0, .., and that the 
demonstrations of the theorems are obtained without difficulty. : 
- 39. Jacobi’s note of the 21st of November, 1838.—Jacobi refers to a 
memoir by Encke in the Berlin ‘ Ephemeris’ for 1837, ‘ tiber die Speciellen 
Storungen,’ where expressions are given for the partial differential coefficients 
of the values in the theory of elliptic motion of the coordinates x, y, z and the 
velocities 2’, y', z! with respect to the elements; and he remarks, Oe if 
Encke’s elements are replaced by a system of elements a, /3, y, a', 3! y' which 
he mentions, connected with — = Encke by equations of a simple form, 
then considering first x, y, z, x',y', 2! as given nates of ¢and the elements, 
and afterwards the elements a, G, y, a’, §', y' as given functions of ¢ and 
2, y, 2, x',y',z', there exists the reinarkable theorem that the thirty-six partial 
differential coefficients = es &e., and the thirty-six partial differential co- 
