338 
PROFESSOR DONKIN ON THE 
with similar equations for In what follows, I shall, as an abridgment, employ 
the symbol E to denote the operation and in like manner I shall put E, for 
-^+^5 so that 
tt'CTy 
Ea=^+^, &c (99.) 
as d'rs ' 
Since r and when expressed in terms of the elements and do not contain <, v, 
we have 
(/n dcosx dfl dn dcos^ 
dv cfcos;^ dv di dcosx d\ 
and 
d cos% 
dM 
=-(>+f) 
\dcosx 
dvi 
d cosx 
} dd 
dv 
dS, 
+ sin / sin sin (^ — y— i') sin {0—v—v,) sin 
1 n djM 
Vtilucs or ~3 
m 
tions (98.). In like manner we find 
in which expression the values of ^ are to be obtained by differentiating the equa- 
VvV CvV ^ 
dcosx di) dcosx du^ dcosx 
dt di d^ di dd/ 
+ sin (6—v—v) sin (^,—y/— *';)(— sin / cos cos t sin cos (*';—«')}. 
Analogous expressions may be found for 
^ tvr y (cl I 
85. Hitherto we have assumed nothing concerning the motion of the axes of 
coordinates. Let us now however take as a first assumption that the plane of xy shall 
always pass through the line of nodes. This implies the conditions* 
v^=v, y=0, y^=0, 
and consequently I=i^— /. 
Introducing these conditions in the expressions at the end of the last article, and 
* The legitimacy of the following processes will be apparent to the reader who shall have followed the 
general reasoning of former articles, though probably not to others. In either case it may be useful here 
briefly to recapitulate the principles now to be applied. The results of art. 79, and in particular the expressions 
(94.), (95.), were established independently of any assumption as to the values of coq, Wj, which are perfectly 
arbitrary. We are therefore at liberty to assume that Wq, are such functions of t that any three func- 
tions of the variables shall constantly =0. Thus the first assumption made in the text is that v^ — v shall con- 
stantly =0. But since such assumptions may cause (as in this case) certain elements to disappear from the 
expression of fi, it is necessary to perform the diflPerentiations of SI with respect to such elements ; the dif- 
ferential coefficients may or may not vanish, on afterwards introducing the assumptions ; if not, we find ex- 
prcssions for them in terms of differential coefficients with respect to other elements which do not disappear ; so 
that instead of differentiations which ought to be performed before the assumed conditions are introduced, we 
have finally only such as may be performed afterwards. The expression “ disappear” does not, it must be 
oljserved, necessarily mean the same as “ vanish.” Thus the condition — j/ = 0, causes v and both to dis- 
appear from ti, but does not of course imply that they both vanish. 
