398 PROCEEDINGS OF THE AMERICAN ACADEMY 



INVERSION. • 



A vid, which differs from unity, but of which the square is equal to 

 unity, may be called a vid of inversion. For such a vid when applied 

 to some other combination transforms it ; but, whatever the transforma- 

 tion, a repetition of the application restores the combination to its 

 primitive form. A very general form of a vid of inversion is 



{A: A) ± {B:B)± (C:C) ± Sec, 



in which each doubtful sign corresponds to two cases, except that at 

 least one of the signs must be negative. The negative of unity might 

 also be regarded as a symbol of inversion, but cannot take the place 

 of an independent vid. Besides the above vids of inversion, others 

 may be formed by adding to either of them a vid consisting of two 

 different letters, which correspond to two of the one-lettered vids 

 of different signs; and this additional vid may have any numei'ical 

 coefficient whatever. Thus 



{A:A) + {B:B)-(C:C)-^x(A:C)Jry(^--C) 



is a vid of inversion. 



The new vid which Professor Clifford has introduced into his 

 biquaternions is a vid of inversion. 



SEMI-INVERSION. 



A vid of which the square is a vid of inversion, is a vid of semi- 

 inversion. A very general form of a vid of semi-inversion is 



{A:A)±{B:B)±sJ-l{C:C)±&c. 



in which one or more of the terms {A: A), (B: B),Scc., have \J — 1 for 

 a coefficient. The combination 



{A:A) ±^-l(B:B) + l{>n±^/-\) (A : B) 



is also a vid of semi-inversion. "With the exception of unity, all the 

 vids of Hamilton's quaternions are vids of semi-inversion. 



THE USE OF COMMUTATIVE ALGEBRAS. 



Cornmutative algebras are espeaially applicable to the integration 

 of differential equations of the first degree with constant coefficients. 

 If i, j, k, &c., are the vids of such an algebra, while x, y, z, &c., 

 are independent variables, it is easy to show that a solution may 

 have the form F {xi -\- yj -\- "^■^ -\- &c.), in which F is an arbitrary 



