511 



a, oil 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 July, 1845, from a transformation of 

 that vector function, which may now be written thus : 



a-'(-a^)-*=^^^"^; (10) 



ana — da a 



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 



da = rd^, a = Srd<; (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^)~^; v.aT= —v.Ta=:l{aT-Ta) ; (12) 



the equation (2) of the internal motion of a binary system be- 

 comes 



d. = 3^^; (13) 



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 

 £, coplanar with a and r, gives the laiv of the circular hodo- 

 graph, under the symbolical form 



_ M(£-IjS^d O 



