5 sums for the system ; t 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 two 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 differential equation may be thus written, 



a being the vector of the attracted body, drawn from the at- 

 tracting one ; t the time ; d the mark of differentiation ; and 

 M the attracting mass, or the sum of the two such masses. 

 This equation gives 



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, 



ada da a r, ^ri n /q\ 



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), 



ai3 + /3a = 0, (4) 



implying that the variable vector a is perpendicular to the 

 constant vector /3 ; and also 



S(a.da - da.a) = 2/3 {t - Q, (5) 



if ^0 ho the value of t at the commencement of the integral. 



