188 REV. M. O’BRIEN ON SYMBOLIC FORMS DERIVED FROM THE CONCEPTION 
dhx 
Hence we immediately find the required expression for the efiective force by 
simple differentiation, as follows, 
du dr , do. dr , ^ 
dt dt'^~^' dt dt 
(Pu d‘^r dr da. d{rM 
dP — dP^'^ 
dt dt 
dt (^+^^dt’ 
^ o\ I /dr , d{rai)\ , 
or effective force zzzl-^—ra Ip+ro/^yy, 
Hence it appears that the effective force is equivalent to the well-known expressions 
along and perpendicular to the radius vector, together with a third part, ruJ per- 
pendicular to the plane of the orbit. 
(83.) The use of the forms u.v and ux v exemplified in the case of motion about a 
centre offeree varying as r~^. — This case appears to me to afford so good an illustra- 
tion of the use of these forms, that I shall give it here briefly. Assuming a, (3, 7, &>, r 
as in the preceding article, it is clear that the symbol of the central force is 
— ^a, and therefore we have 
(Pu (X 
lp~~p^’ 
whence 
therefore (by art. 81), 
u , 
dPu 
dP 
dA 
dt 
= 0, since u—ra, and a.a=0. 
= 0 . 
( 1 -) 
( 2 .) 
This indicates that there is no variation of A, and consequently (see art. 81) that 
the plane of the orbit, and the rate of description of area is constant. 
If we put for A its value (art. 81) \u.^ and for u and ^ their values ra and -^5 
observing that ^=<yj3, we find 
or, by (2.), 
A=^ra. ^Ja-l-r«/3)=^r"a;a.^, 
A=|/ia./3 {h=r^a, as usual). 
Now it is singular that (1.) admits of immediate integration, instead of requiring the 
well-known transformations : for, in it, put for a its value, — “ ^ (see art. 82), and 
we have 
d^u fJt' d^ fj. d^ ^ 
dp r'^to dt h dt ’ 
du M. „ . 
di=Tfi+ constant. 
wherefore, integrating, we find 
