308 
PROFESSOR DONKIN ON THE 
expressions are obtained by Pontecoulant*, will show the convenience of using the 
coefficients c,} instead of [c^, cj (in Ponte'coulant’s notation (c;, Cy)). 
[Jt will be observed that the forinulse for (g)', ts', / at the end of the last article, 
do not agree with those of Pontecoulant (p. 330) for the variations of the corre- 
sponding quantities a, co, (p. The reason of this is as follows : — In Pontecoulant’s 
notation (p expresses the same as / in this paper, and a the same as v. But a (the 
longitude of the periheiion) is not the same as w; the former being measured entirely 
in the plane of the orbit from a radius yeciov, fixed in that plane and assumed as 
the origin of longitudes. Consequently g, in Pontecoulant (which we will call for 
distinction), is not the same as (g) in the present paper. In fact, if we equate the 
expressions for the mean anomaly in the two notations, we have 
g^ — co= (g) — zs ; 
also it is evident that if we put /3 for the angle between the node and the origin from 
whieh 0 ) is measured, we have — co?,idv, and 7z~v-\-(5-{-co, so that 
dz) = doij-]-{l — cos i)dv. 
If then it were allowable to consider R as capable of being expressed as a function of 
u and g^ instead of w and (g), and if we represented by (R) the expression for Pt so 
transformed, we should have 
c/R 
^/(R) 
f/(R) 
^rf=,+^rf(s)+&c.=^</»+^rfs,+&c.; 
and if, in the two first terms, we put for dzj and d{^) the values d'UT—da-\-{\—co^i)dv, 
d{z) = dz^-\-{\ — cm t)dv, and compare the two expressions, we find 
</R 6?(R) c?R ft’(R) 
d^s dcj) ’ d{i) ds^ ’ 
These relations, together with the equation 
zd — cJ ( 1 — COS djf 
are easily seen to render the expressions at the end of art. 55 identical with those of 
Ponte'coulant ; in fact, it is by an equivalent transformation that the latter are 
finally obtained by that author from the correct expressions in p. 328. But it is to 
be observed that this proceeding is founded upon a false assimiption ; for it is im- 
possible to express R as a function of a, e, /, t', gp oj, as is obvious from the considera- 
tion that Pt, in its original form, is not a function of (s) — ^ merely, but also of (g) ; 
whilst (g) is not expressible as a function of the new elements, as is shown by the equa- 
tion d{i) = dz^-\-{\—QO'$,i)dvf It would be out of place to enter further into this sub- 
* Theorie Anal, clu Syst^me du Monde, tome i. pp. 316-330. 
t On the meaning of this expression, see below, art. 73. 
X It would be a work of some trouble to trace accurately the process by which Laplace arrives at the for- 
