MINDING’S THEOREM. 677 
be expressed by a special linear and vector function, x, which possesses the 
following characteristic properties ; 
SxaxB=SaB , 
(of which a particular case is 
Tya=Ta,) 
and ; VyaxB=xVaB. 
’ 
Also the conjugate of x is its reciprocal, or 
=X 
These premised, we may attack the question. 
2. When any number of forces act on a rigid system; (, at the point a,, 
B, at a, &c., their resultant consists of the single force 
acting at the origin, and the couple 
c=—Vieax : - : < 4 (1) 
If these can be reduced to a single force, the equation of the line in which 
the force acts is evidently 
Weo=> Vaan eo , : (5) 
Now suppose the system of forces to turn about, preserving their mag- 
nitudes, their points of application, and their mutual inclinations, then 
Minpine’s Theorem, proved (in CRELLE’s “Journal,” vols. xiv., xv.) by an 
excessively elaborate process, assigns certain fixed curves in space, each of 
which is intersected by the line (5) in every one of the infinite number of its 
positions. 
3. To prove this, and to find the curves in question, we may proceed as 
follows :— 
Operating on (5) by V.8, it becomes 
pB?—BSBp=4B—-4'B 
