DIFFERENTIAL EQUATIONS OF DYNAMICS, ETC. 
309 
ject here, especially as some of the most important principles involved in it have 
been discussed elsewhere See also Appendix B.] 
57 . Returning' to the expression (69.), art. 54, it may be observed that the coeffi- 
cients {c„ Cj] are to be expressed in terms of Cj, &c., and this involves no difficulty 
when each of the two sets of elements c^, &c., a^, &c. can be expressed in terms of the 
other explicitly, as was the case in the example just discussed. Suppose, however, 
that the normal set a,, &c., &c. are given in terms of the set Cj, &c., but that it is 
impracticable or inconvenient to obtain the converse equations expressing the latter 
in terms of the former. In this case we may proceed as follows. 
Adopting the notation of art. l-i-, and putting/’, g for any two of the set Ci, Cg, &c., 
we have 
d{f,g) . 
^d[bi, tti) ’ 
suppose this equation written at length, and then, after multiplying by 
d{bj, aj) 
d{f,g) 
, let 
each side be summed with respect to all binary combinations fg. The result is (see 
art. 1, equation (4.)), 

(the summation referring to the combinations f, g). Again, if the former equation 
di 7) Cl] 
be multiplied by where p, q represent any two of the normal elements, a^, &c. 
&c., except a conjugate pair, and the sum be taken as before, we have 
The two formulae (70.), (71.) give n{2n—\) linear equations for determining the 
«(2w— 1) unknown quantities {/*, g}; the coefficients of the latter being all given 
functions of Ci, &c. But such cases will hardly occur in practice. (With respect to 
the form of the above system of linear equations, it is easy to show that the complete 
determinant of the coefficients is =1.) 
58. The integration of the formulae (65.), art. 53, would give the means of ex- 
pressing the solution of the system 
,_(m _ dW 
di/i ’ dxi 
mulse alluded to in a preceding note, as the various steps of it are to be found in different places, the notation 
is somewhat inconsistent, and the results do not profess to be rigorous. My impression is, however, that 
Laplace nowhere commits the fallacy of assuming (for example) that R is a function of r, v, z, or r, v, s (see 
vol. i. p. 295), where v is the angle described by the radius vector on the varying plane of the orbit. 
* See Jacobi’s two letters to Professor Hansen in Crelle’s Journal, vol. xlii. 
+ d(u, v) , , . . du dv du dv 
T t. e. using ( as an abbreviation for 
d{x, y) dx dy dy dx 
2 T 2 
