XXXVili 
= sums for the system; ¢ is the time, d the characteristic of 
differentiation ; A (where used) is the mark of finite diffe- 
rencing. 
To illustrate the method of treating equations of such forms 
as these, let us consider briefly the problem of éwo bodies, or 
of one body, as it presents itself, in the method of quaternions, 
with Newton’s law of attraction, coordinates being not em- 
ployed. The differentia] equation may be thus written, 
d? M 
ae PST! (1) 
dt ay (— a’) 
a being the vector of the attracted body, drawn from the at- 
tracting one; ¢ the time; d the mark of differentiation; and 
M the attracting mass, or the sum of the two such masses. 
This equation gives 
aa” “ta 
aa ap a = 0; 2 
eapctgas (2) 
which expresses merely that the force is central; and gives by 
integration a result already alluded to (as independent of that 
function of the distance which enters into the law of attrac- 
tion), namely, 
a da da a 
3d ~ diz Pi P= 9 (3) 
the constant 3 being a new vector, perpendicular in direction 
to the plane of the orbit, and in magnitude representing the 
double of the areal velocity, which velocity is thus seen to be 
constant, as also is the plane. For we have at once, by (3), 
af + Ba = (4) 
implying that the variable vector a is perpendicular to the 
constant vector 3; and also 
{(a.da — da. a) = 2/3 (¢ — &); (5) 
if t) be the value of ¢ at the commencement of the integral. 
