Bam. — Contrilmlionn to the Tlieori/ nf Scrctvs. 61 



To verify C = we may take the group of terms involving fi^ ; they are 



+ S/usXjXiFAiAj + (S/ujXiAa f'^AoX4 + Sfi^K^iVX^i. 



If we substitute for Xi the expression oX, + IX^ + cXi where «, h, c are scalars, 

 this expression is seen to vanish identically. In like manner for the terms 

 involving ^,, fj., ^3, ut- 



The last identity Z> = A (97) is somewhat remarkable. If we take the terms 

 only involving n^ in X, we have 



X = VfjifJsSXiXiX, + VfiiHiSX^XiX^ - F^s/us^^XjA, - Fjus/jjiSXiXiXs. 



The terms involving jus in I) are in number 12, of which three are 



VXiX^SXi/ii/jis + VXtXiSX^iiifis + F^XaXi&'XjjUjjus; 



but this is equal to F/UjiuSXiXoXj, because from a known vector formula 



00X1X2X3 = K X1X2 . 5X30 + J^XjXa . 'S'XjO + F^XsXj/SXjfi. 



Thus we show that each term in X equals the sum of three terms in D ; 

 and the verification is complete. 



VII. — Ecpresentation of Screiv Systems of the third order hy Linear Vector 



Functions. 



There is, perhaps, no part of the theory of quaternions of greater interest 

 to the student of mathematical physics than the theory of linear vector 

 functions introduced by Sir William Hamilton.* This beautiful theory 

 exhibits in the most lucid manner the geometrical element common to 

 many investigations in varied branches of mathematical inquiry. It is 

 known that the strain of an elastic body displaces any vector of that body 

 into another vector which is a linear vector function of the original vector. 

 It is also known that the vector expressing the impulsive moment applied 

 to a rigid body free to move about a point generates a twist velocity 

 which is a linear vector function of the original impulsive moment. These 

 are elementary applications ; and, as instances of more recondite uses of the 

 linear vector function, we may mention its employment by the late Pi-ofessov 

 Willard Gibbs, and, more recently, in the important investigations of Professor 

 Conway in molecular mechanics. 



Of course, to speak strictly, the theory of linear vector functions does not 



* "Plemente," vol. i., p. 486. 



