200 Application of Quaternions to the Theory of Rotation. 



To verify this a posteriori, if in the first place we substitute 

 for x l its value M, 2 . xx', we have 



rfr _iVJiX * + m; dt * v 



and thence 



-r 1 A, + *r — r 1 x, = MV -r-A. 



dt lT M, (/< x rf/ 



Also 



rfA, . / l dU, A „ 1( rfA\ 4 1 «ZM, , 



— ^A^^rl-r^ rr A, + M,A'— - )A, = ^ j-JA 



eft J \M, <// in ^ J <#/ ' Mj dt 



+ M 2 A'^A'A 



'J 



whicli reduces the equation to 



( A » + x i) + M i • A Sr A A=M i ^ it A « 



Or observing that 



A 1 2 + x 1 = 2A 1 = 2M 1 A'A, 



and omitting the factor A from the resulting equation, 



2 dM, A , A ,rfA A , ,<7A 



TTjTo— jAa' + A'— A' = x' — . 



M, 2 A rf* cfif 



Or since 



Substituting and dividing by A' 



„ ( , dX J da , dv\ . . , , dA dA , 

 \ dt r dt ^ dt/ dt dt 



Or finally, 



-(<|.*/i+*l)( i +«v+*'+^ 



=-(w+>' + «)(^ + i|+^) 



which is obviously true. 



58 Chancery Lane, 

 July 1, 1848. 



