DIFFERENTIAL EQUATIONS OF DYNAMICS, ETC. 
349 
different manner*. The equations of art. 93. are merely transformations of the 
original differential equations of the problem, without any integrations ; they are 
however in a form which might perhaps be used advantageously in certain cases for 
the purposes of physical astronomy. Those of Jacobi are obtained by employing all 
the four usual integrals of the problem, and are shown to include an additional inte- 
gration. They have however the disadvantage of substituting the coordinates of two 
fictitious bodies for those of the actual planets, and would probably be inconvenient 
for ordinary practical use ; though in a theoretical point of view they seem to deserve 
more attention than they have hitherto received. It would be wrong to take leave 
of this celebrated problem without referring to another transformation by M. Ber- 
TRAND-i-, which, as has been remarked by a recent writer in the same journal, effects 
six integrations, and therefore represents the furthest advance which has yet been 
made towards a rigorous solution. 
Appendix A. 
When the method described in Theorem VII. (art. 49.) is applied to the solution 
of a system of equations of the form (I.), of which n integrals, ... satisfying 
the conditions [«;, a,]=0, are given, the first step is to express the n-\- \ partial dif- 
ferential coefficients &c., and namely, the values of y^, •••yn and — Z in 
terms of x^, ... «!, ... and t. The direct process is then to find X by integrating 
the expression yxdx^-\-y^dx^-\- ..-\-yJiXn—7jdt, and afterwards to form the remaining 
integral equations ^ — ; when this process is adopted, the inferior limits in 
the integrations are perfectly arbitrary ; in other words, we may add to X an arbitrary 
function of aj, «25 ••• without altering any of the general properties of tlie final 
system of integrals. 
But it is generally much more convenient to perform the differentiations with 
respect to a^, ...a^ first, and integrate afterwards; thus we obtain the remaining 
equations in the form 
When this plan is followed, the limits are still arbitrary if it be only required that 
the equations thus obtained shall be true-, but if it be required that they shall form, 
with the given integrals, a normal solution, it is necessary to take the limits in such a 
manner that the functions equated to b^, h^, ... shall be the partial differential coef- 
ficients with respect to a^, a^, of one and the same function-, which will not 
generally be the case unless care be taken that it should be so. 
In practice, the expression for c?X usually consists of several terms, of which each 
* Comptes Rendus, 1842, part 2. p. 236, &c. 
3 A 2 
t Liouville’s Journal, 1852. 
