514 



if we make, for abridgment, 



0„,».=: »^„. n' (/3a)™ (ai3)"'a-X-a^)-*-«-"' (23) 



where 



_ 1.3..(2n-l) 3.5..(2n-+l) 



^"'"'- 2.4.. (2w) ^ 2.4.. (2»0 • '^ ^ 



The attraction 0(j3 + a) which a mass-unit, situated at the 

 beginning of the vector /3 + 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, <pn, n', of which 

 the intensities are determined by the tensors, and of which the 

 directions are determined by the versors, of the expressions 

 included in the formula (23) ; or by the following expressions, 

 which are derived from it by the rules of the calculus of qua- 

 ternions : 



T<p„,n' = mn,n' ( T ^ j (Tq)""^ ; (25) 



v<f>n,n' = (U . /3a)»-' (ua)-i = (u 3^)"'"'u ( " 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 + j3 ; then the positive or negative rotation in the same 

 plane, from the same direction of — a, to the direction of the 

 component force ^n, n'> is expressed as follows : 



angle, from— a to force 0„_ „/, zz{n — n')c ; (27) 



and 

 intensity of same component forcer m„,,i'(-j a~^ (28) 



The case n- o, n' - o, answers to the old tractor ^(a), or to a 

 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 j3 shall be con- 



