15 



Academy for July 20, 1846, the following equation of the 

 ellipsoid (there numbered 44), 



T(ip + pK)^K'-L\ ■ (1) 



was given, as a transformation of this other equation of the 

 same surface (there marked 35) : 



T{ap + pa + (3p-pli) = \', (2) 



which was itself deduced by transforming, according to the 

 rules of quaternions, the formula 



(ap + pay-(i5p-p(5y-=l; (3) 



this last quaternion form of the equation of the ellipsoid having 

 been previously exhibited to the Academy, at its meeting of 

 December 8, 1845. (See the equation numbered 21, in the 

 Proceedings of that date.) The symbols a, (5, denote two 

 constant vectors ; the symbols t, k, denote two other constant 

 vectors, connected with them by the relations 



a+/3 = -2 2, a-/3 = -^ ^, (4) 



t — K I —K 



where i' - k^ is a negative scalar ; and p denotes a variable 

 vector, drawn from the centre to the surface of the ellipsoid : 

 while T is the characteristic of the operation of taking the 

 tensor of a quaternion. 



If a new variable vector v be defined, as a function of the 

 three vectors t, k, p, by the equation 



(k^ - l^y V = {k^+ l^) p + ipK + Kpl, (6) 



it results from the general rules of this calculus that this new 

 vector V will satisfy each of the two following equations : 



S.vp = l; S.vdp = 0; (6) 



which give also these two other equations, of the same kind 

 with them, and differing only by the interchange of the two 

 symbols p and v : 



S.|ov=l; S.(odi. = 0; (7) 



where d is the characteristic of differentiation, and S is that 



