350 
PROFESSOR DONKIN ON THE 
contains one of the variables only. Suppose one of these terms is 
(p{x, a^, ... a„)dx, 
so that, so far as this term is concerned, we have 
X=^<p{x, a„ flg, ... a„)dx, 
where A is an arbitrary function of a^, ... a„. Consequently 
and we see that we should not In general obtain the differential coefficients with 
respect to a^, ... of one and the same function X, by merely integrating- p-, &c., 
(JjH ^ Utl^ 
with respect to x, from the same inferior limit A, chosen at ha,zard. 
But it is evident that we shall attain this end if we adopt the following- simple 
rule : — ■ 
Integrate &c. with respect to x, taking the same inferior limit in each case, 
namely, either 
(1) a value A of zr which satisfies the equation (p(x, ... a„) = 0, or 
(2) any determinate constant (i.e. not a function of aj, ... «„). 
For example, in the problem of central forces (Part I., art. 28, &c.), we had (see art. 19.) 
dX= —hdt-\-cd6-\-(2m{h-\-(p{r)) — lfr~^^^dr-\- (P— sec^ X)^dX 
(where r, 0, X are the three variables). 
The very troublesome process of differentiating X with respect to h, k and c after 
first finding X by integrating the above expression, is avoided by the method adopted 
in art. 29 ; namely, by differentiating- first, and integrating afterwards. In the inte- 
grations with respect to r, the inferior limit is one of the roots of the equation 
2m{Ji-\-(p(r)) — k^r~^=0. 
namely (in the case of elliptic motion), the perihelion distance ; and in those with 
respect to X, the inferior limit is 0 ; so that the rule above given is observed. 
At the time of writing the article referred to, neither the rule itself, nor the neces- 
sity of attending to the limits, had occurred to me ; it was therefore, strictly speak- 
ing, accidental that the final integrals were obtained in a normal form. 
In treating the problem of rotation (Section III.), I perceived the necessity of cau- 
tion as to the limits, if the former order of proceeding were adopted ; but preferred 
avoiding the risk of error altogether, by performing the integrations first, so as to 
obtain the actual expression for V. The final equations (R.), art. 44, might however 
be obtained in a more simple way by differentiating^r.^^ ; thus we should have (see 
equations (45.), (46.)), observing tbat^=^j &c. (art. 44.), and putting 
