30 Sir W. Rowan Hamilton on Quaternions. 



2S.a = «-« = 0; 2V.« = « + a=2«; 



2S.«a' = ««' + «'«; 2 V.aa' = aa' — a'a; 



2 S. aa'a" = aa'a" — a"a'a ; 2 V.aa' a" = aa'a" -f a" a'a ; 



&c. &c. 



of which the law is evident. 



21. The fundamental rules of multiplication in this calculus 

 give, in the recent notation, for the scalar and vector parts of 

 the product of any two vectors, the expressions, 



S.aa' = — (xx' + yy' + zz') ; 

 V.aa! = i(yz' — zy 1 ) +j(zx' — xz') -f k(xy' —yx 3 ) ; 

 if we make 



a = ix +jy + kz } a! = ix' +jy' + kz', 



x,y,z and x\ y\ z' being real and rectangular co-ordinates, 

 while i,j, fc are the original imaginary units of this theory. 

 The geometrical meanings of the symbols S.aa', V.aa', are 

 therefore fully known. The former of these two symbols 

 will be found to have an intimate connexion with the theory 

 of reciprocal polars; as may be expected, if it be observed 

 that the equation 



S. a a! — — a 2 



expresses that with reference to the sphere of which the equation 

 is 



a 2 = — a 2 , 



that is, with reference to the sphere of which the centre is at 

 the origin of vectors, and of which the radius has its length 

 denoted by a, the vector a' terminates in the polar plane of the 

 point which is the termination of the vector a. The latter of 

 the same two symbols, namely V. a a', denotes, or may be 

 constructed by a straight line, which is in direction perpen- 

 dicular to both the lines denoted by a and a', being also such 

 that the rotation round it from a to a! is positive; and bear- 

 ing, in length, to the unit of length, the same ratio which the 

 area of the parallelogram under the two factor lines bears to 

 the unit of area. The volume of the parallelcpipedon under 

 any three coinitial lines, or the sextuple volume of the tetra- 

 hedron of which those lines are conterminous edges, may 

 easily be shown, on the same principles, to be equal to tlie 

 scalar of the product of the three vectors corresponding; this 

 scalar S.aa' a", which is equal to S(V. aa' . a f '), being positive 

 or negative according as a" makes an obtuse or an acute angle 

 with V. a a', that is, according as the rotation round a" from 

 a' towards a is positive or negative. To express that two 



