G8] ON THE APPLICATION OF QUATERNIONS TO THE THEORY OF ROTATION. 409 
but I have not ascertained whether this formula leads to any results of importance. 
It may, however, be made use of to deduce the following property of quaternions, viz. 
if A 1 — ilfjA'A, M 1 as before, then 
1 
Ki 
d-A x 
dt 
A j + 
1 /_ d A . d/c\ 
A 2 M A+ Tt) 
in which the coefficients of A' are considered constant. 
To verify this a posteriori, if in the first place we substitute for k x its value 
M f/e/d, we have 
d/Ci _ M 2 / d/c 2 dM 1 
dt ~ K Tt + M] ~dt Kl ’ 
and thence 
Also 
dAi . 1 dM x -.j. , dA . 
dt k ' + v t ~d* *-* '-&*■ 
dA2 . /1 CLlHi . n/r \ / CLi\\ , I Hum.2 . „ 7ii-„ a / w-n. . , . 
— ( U ~~J7 + M X A ) Aj = T/f -jj- A,- + M x A A A, 
1 dMi 
M, dt 
/ dA 
dt 
1 dil/j A „ 
ilfi d.t 
/ dA A , 
dt 
which reduces the equation to 
1 dd/i 7i / o a / dA , ,j dA 
1/2 di (Al " + ^ + i/r A It AA ~ MfKl dt A ’ 
Hence observing that 
A1 2 + *i = 2Aj = 2M 1 A , A, 
and omitting the factor A from the resulting equation, 
2 dM 
M? dt A/ + A ' T - A ' ~ K 
, dA ., _ , dA 
dt A ~ K ~dt ; 
or since 
jÿj- = 1 — AA' — fifj! — vv , 
substituting and dividing by A', we obtain 
0 (\ r d^ 1 /d/A / diA , , dA dA , 
2 Y di +li Tt +v dt) =KA dt~di A ' 
or finally, 
, dA , dp , dv 
2 ( v Tt + ^ Tt + is ) = - ,v - fo ') (• 7Z + j % + * is 
d£ ,y d£ di 
dt dt. 
(* d£ 1 J dt 1 ,v dt, 
■ - (*v +>' + foo f'5+js +i â)-f'î + iî + *is) < tV +>'+*»'). 
_ (* is U is+ k it) + a ' +>' + fo '> 
which is obviously true. 
C. 
52