511 
a, on another mass-unit situated at the end of that vector, must 
be important in the theory of the Moon; and generally in 
the investigation, by quaternions, of the mathematical conse- 
quences of the Newtonian Law of Attraction. The integration 
of the equation of motion (2) of a binary system was deduced, 
in the communication of J uly, 1845, from a transformation of 
that vector function, which may now be written thus: 
a-"(—q?)*= Fe A at (10) 
where d is, as in former equations, the characteristic of diffe- 
rentiation. And the hodographic theory of the motion of a 
system of bodies, attracting each other according to the same 
Newtonian law, so far as it was symbolically stated to the 
Academy, at the meeting of the 14th of December, 1846, de- 
pends essentially on the same transformation. In fact, if we 
make 
ada—daa 
da=rdt, a=Srdé; (11) 
and if, by the use of notations explained in former communica- 
tions, we employ the letters u and v as the characteristics of 
the operations of taking the versor and the vector of a quater- 
nion, writing, therefore, 
U(a) =a(—a’)-4; Veat=—V.ra=}(ar—ra); (12) 
the equation (2) of the internal motion of a binary system be- 
comes | 6rd 
—mdu((rdé) 
mite yD 13 
v(r(rdé) ) 
where the denominator in the second member is constant, by 
the law of the equable description of areas. Hence, this second 
member, like the first, is an exact differential; and an imme- 
diate integration, introducing an arbitrary but constant vector 
é, coplanar with a and 7, gives the law of the: circular hodo- 
graph, under the symbolical form 
_ M(e—vGrdé) | 
Aas V.r\rdt . (14) 
dr= 
