514 
if we make, for abridgment, 
n,n = Mn, w (Ba)" (aB)"a-'"(— a?) (23) 
where 
1.3..(2n—1) _ 3.5..(2n’+1) 
2.4.. (Qn) ~~ 24..Qn) * (24) 
Mn, nv = 
The attraction ¢((3 +a) which a mass-unit, situated at the 
beginning of the vector +a, exerts on another mass-unit 
situated at the end of that vector, is thus decomposed into an 
infinite but convergent series of other forces, gn, n, of which 
the intensities are determined by the fensors, and of which the 
directions are determined by the versors, of the expressions 
included in the formuia (23) ; or by the following expressions, 
which are derived from it by the rules of the calculus of qua- 
ternions : 
pay = Mau (TB) (ra) (25) 
: " a 
vba =(-Ba)'-" (vayt=(vB)""o (a). (26) 
Let a, b, be the lengths (or tensors) of the vectors a, 3, 
and let c be the angle between them, which angle we may 
conceive to express the amount of the positive rotation, in 
their common plane, from the direction of —a to the direction 
of +; then the positive or negative rotation in the same 
plane, from the same direction of —a, to the direction of the 
component force gn, n, is expressed as follows : 
angle, from—a to force on, 7, =(n—M’)C ; (27) 
and 
‘ : ; b n+n’ 
intensity of same component force = mn,» (;) ay (28) 
The case n= 0, n’= 0, answers to the old tractor ¢(a), or toa 
force of which the intensity is represented by a~*, while its 
direction is the same as that of —a. 
IV. Thus, if the vector a be conceived to begin at a point 
-B, and to end at the point c, while the vector 8 shall be con- 
—_ 
