\ 



41 



quaternion (3) vanishes ; so that we may write for this case 

 the equation, 



= AXSF. 0/3-7-2/3); (8) 



which agrees with the equation (19) of the abstract of Decem- 

 ber, 1845, and in which the second member is in general a 

 scalar symbol, denoted lately by w, and having the significa- 

 tion already assigned. When the resultant force vanishes, with- 

 out the resultant couple vanishing, then the denominator or 

 divisor 2/3 becomes null, in the fraction or quotient (3), while 

 the numerator or dividend, 2F. a/3, continues different from 

 zero ; and when both force and couple vanish, we fall back on 

 the equations (18) of the former abstract just cited, or on those 

 marked (2) in the present communication, as the conditions of 

 equilibrium of a free but rigid system. Finally, the scalar 

 symbol 



c = - 25.0/3, (9) 



which enters with its sign changed into the second member of 

 . the formula (1), and which, when the resultant 2/3 of the 

 forces /3 vanishes, receives a value independent of the assumed 

 origin of the vectors a, has also a simple signification ; for 

 (according to a remark which was made on a former occasion), 

 there appears to be a propriety in regarding this scalar symbol 

 c, or the negative of the sum of the scalar parts of all the 

 quaternion products of the form a(5, as an expression which 

 denotes the total tension of the system. In the foregoing 

 formulae the letters 5 and V are used as characteristics of the 

 operations of taking respectively the scalar and the vector, 

 considered as the two parts of any quaternion expression ; 

 which parts may still be sometimes called the (algebraically) 

 real and (algebraically) imaginary parts of that expression, 

 but of which both are always, in this theory, entirely and 

 easily interpretahle : and in like manner, in the remainder of 

 this Abstract, the letters Tand L' shall indicate, where they 

 occur, the operations of taking separately the tensor and the 



