40 



which have already been often adduced and exemplified by 

 him, in connexion with other geometrical and physical re- 

 searches ; then the four constituent numbers, w, x, y, z, of 

 this quaternion (3), will have, in the present question, the 

 meanings which we are about to state. The algebraically real 

 or scalar part of the quaternion (3), namely, the number 



w = S{-S.r.a(5-i-'2,(5), (5) 



which is independent of the imaginary or symbolic coefBcients 

 i,J, k, will denote the (real) quotient which might be otherwise 

 obtained by dividing the moinerd of the principal resultant 

 couple by the intensity of the resultant force ; with the known 

 direction of which force the axis oi ih\% principal {an A known) 

 couple coincides, being the line which is known by the name 

 of the central axis of the system. And the three other numeri- 

 cal constituents of the same quaternion (3), namely, the three 

 real numbers .^', y, z, which are multiplied respectively by those 

 symbolic coefficients i,j, k, in the algebraically imaginary or 

 vector part of that quaternion, namely, in the part 



iv +jy + kz= F(S V. a/3 -- 2^), (6) 



are the three real and rectangular co-ordinates of the foot of 

 the perpendicular let fall from the assumed origin (of vectors 

 or of co-ordinates) on the central axis of the system. These 

 co-ordinates vanish, if the origin be taken on that axis ; and 

 then the direction of the resultant force coincides with that of 

 the axis of the resultant couple : a coincidence of which the 

 condition may accordingly be expressed, in the notation of this 

 Calculus, by the formula 



0=F(sr.«|3-2/3); (7) 



whereas the second member of this formula (7) is in general a 

 vector-symbol, which denotes, in length and in direction, the 

 perpendicular let fall as above. In the case where it is pos- 

 sible to reduce the system of forces to a single resultant force, 

 unaccompanied by any couple, the 'scalar part of the same 



