JoLY — The Associative Algebra appUcahle to Hyperspace. 107 



44. Though, intending to return to a special class of operators of 

 the type ^( )^"\ I shall now supplement Clifford's Paper "On the 

 Free Motion under no Forces of a Rigid System in an w-fold Homaloid " 

 hy a few remarks. 



By Newton's law, if ^i is the impulsive force applied to an element 

 of mass m^ at the extremity of pi, the momentum generated (nhpi) is 

 equal to the impulse, or mipi = ii. 



From this we may derive the two following equations, on multi- 

 plying by pi and pi, 



««iPiPi = Pi^i, and Mipi^ = pifi. 



Now, if nil is part of a system, on summation over the entire 

 system, we find 



S^ipi = '^ii, %nhp-^p-^ = Spi^i, and Swhpi" = Spifi- 

 Considering two elements of the system, m^ and nh, the impulses 

 ^1 and ^2 may be written more explicitly in the forms ^'i + ^n, and 

 ^'2 + lai, respectively, where ^12 is the impulse on mj arising from its 

 connexion with m^, and |^2i is the impulse on m^ arising from its con- 

 nexion with m^. If these are equal and opposite they cancel in ^^1, 

 and %^i is then equal to the sum of all the external impulses acting 

 on the system. 



Again, Spi^i = Spil'i + 5 (pifi2 + ^2^21)- 



Consider the term 



(pi^i2 + p^izi) = (pi - P2) fi2 (as fi2 + ^21 = 0) ; 



^2(pi - P2) f 12 will vanish if the mutual impulse acts along the line 

 between the elements ; but S (pi - ps) ^12 will not vanish, unless the 

 mutual impulse acts at right angles to that line. On the assumption 

 that the mutual impulse acts along the line joining the elements, 

 ^1 may be taken as the external impulse on the element j«i, in the 

 equation 



2»^l V2P1P1 = 5 Vzpi^i ; 



but it cannot so be taken in ^m^Spipi = '^iSpiii, from which the 

 impulses of constraint do not disappear. 

 Next, in the scalar equation 



^m,p,' = ^p,i, = s:$p^ei + js^ (pi - P2) u ' 



the impulses of constraint (acting along pi - P2) will not disappear, 

 unless pi - P2 (the velocity of mi relative to niz) is at right angles to the 

 line joining ^i and m^ (or at right angles to ^12). For a rigid body 

 in the equation ^iUipi^ = '^Spiii, $1 may consequently be taken equal 

 to the external impulse on mi. 



