344 
PROFESSOR DONKIN ON THE 
(where ^ is the distance between m and m,); and in like manner 
A'=— zy,)(^-^— r-^) ; 
we have therefore A'+A'=mm/yz^— zy,)(r^"®— r"^), 
with similar expressions for B'+B;, C'+C; so that, when the common factor 
is omitted, the equation (115.) becomes 
(yz,-zy,)|+(zx^-xz>+(xy-yxX=0, 
which is evidently the equation to the plane containing the two radii vectores. Thus 
the theorem in question is verified. 
90. To return from this digression : the motion of the line of nodes in the principal 
plane will be given by putting in either of the equations (102.), and introducing 
the value of ^/osin v— cos v from (103.). In this way we find, after slight reductions, 
. , , , dO,, 
pPiSm l.v'=p^cos 1 1 -j[-tP cos/-^S 
in which we may substitute for cos/, cos/,, the values given by equations (110.); this 
gives 
(rpp, sin I . = (p^+pp, cos I)^j + (p^-\-pp, cos I)^ ; 
or, if we introduce the actual values of jj-’ we find 
pp/= — mm,rr,sin (O—v ) sin (^,— v) x { /? cos / . (^"^ — r“®) +jB,cos /,(^“® — r ~^) } . 
It is not my purpose however to enter further into details ; and I shall conclude 
this subject by briefly examining the consequences of a slightly different assumption 
as to the motion of the axes of coordinates. I shall suppose, namely, that the plane 
of xp still always coincides with the principal plane, but has a sliding motion such 
that the axis of x always coincides with the line of nodes. 
91. The assumption made at the end of the last article implies the condition »=() ; 
and /y 2 will no longer be 0, but must be determined by equations (102.) ; either of 
these gives (putting v=v=0, and reducing by means of (103.), (109.), &c.) 
;?p,smI.&»2=PyCos/,^+7?cos/-^', (116.) 
which coincides, as of course it ought, with the expression given for v' on the former 
hypothesis (art. 90.). The difference is that Q, Q, now no longer contain v. 
The values of iy,,, /w, are obtained at once by putting v=0 in the equations (112.), 
(] 13.) ; and all the conclusions which were derived independently of any supposition 
as to the value of co.^, subsist as before, when modified by putting v=0. 
We may add one more equation, which is required in forming some of the expres- 
sions for the variations of the elements; namely. 
, I — sml 
tan -= ^ ' T’ 
2 <r -{-p -\-pi cos 1 
( 117 .) 
