Interpretation of Quaternions. 117 



The following cases will exemplify the above interpretation 

 of quaternions. 



If ABC be a spherical triangle, the radius being equal to 

 unity, and if Q, Q', Q" indicate the rotations of the radius 

 vector from B to C, from C to A, from A to B respectively, 

 it is clear that we must always have 



Q"Q'Q = QQ"Q' = Q'QQ" = 1 . 



In order to find the quaternion which will represent the ro- 

 tation from the line (/, m, n) to the line (/', m', n 1 ), we may con- 

 struct a quadrantal triangle such that (I, m, ii)\ (/', m' 9 n 1 ) pass 

 through the angles opposite to the quadrantal sides ; and if Q 

 be the required quaternion, 



{li + mj -f nk) {I'i + mj + n[k) Q = — 1 ; 

 but since 



(U + mj+n1cf = (Vi + mj+n'kf^ - 1, 



{I'i + m'j -f n'k) Q = (li + mj + nk) 



— Q = (I'i + mj+ n'k) {li + mj+ nk) 



Q = --(U' + mm' + nri) + i{mn' — ?nn) +j(nl' -?i'l) + k(l?n , —l l m). 



To find the quaternions which will represent the rotations 

 from the three coordinate axes to the line (/, m, ?i), we need 

 only put in the above equation, 



?n = 0, ?i — 0; n = 0, /=0; 1=0, m = 



in succession ; hence, dropping the accents, 



Q x ——l—jn + km 



Q y =—m + in — kl 



Q z ——n — im-\-jl\ 

 to which may be added the following relations : 



Q/+Q/+Q/=-i 



IQ. + mQy + nQ^-l 



iQ v +jQ v + kQ z -—H —jm — hi 



nQ — mQ z = i— l(il +jm + kn) 



IQ Z — nQ x —j—m{il +jm + hi) 



m Q v ~~ ^Q, =k — n(il +jtn -f- hi) 



(jm - hi) Q x + {in - Id) Q y + (jl- im)Q z = - % 



={nQ y -mQ,f+ (lQ z - n Q t )*+ (»iQ,-/Q; 



