
G. PLARR ON THE ELIMINATION OF a, B, y, ETC. 269 
the terms being grouped into 
da = B(u,da — v,db) + y(vida — vide) , 
we see that indeed da = 0, when 
which corresponds to dp being parallel to <jv, and V(<vdp) = 0. 
Similar remarks relate to d8, dy, with relations similar to (52 dzs). It is 
precisely these particular modes of variation of a, 8, y which work out the ful- 
filment of the conditions of integrability. But we must remember that (52 b7s) 
is unfit for integration, and gives only actual values. 
Having formed the expressions (51) of a,, etc., involving the conditions 
(389), we calculate the sum 
Yo, = 42V7 (adv) = 8( dv)’. 
Having also found by (45) 
VII = 441, 
the expression (35) of S<II may now be used, because its two terms 
Sal =—jV'll + 73’: 
are expressible now both in function of the same quantity (<1v)’, as 
(58) Yo? = 5V'll = 8( dv)". 
§ 4. Connections between the preceding method and the method founded on the representation of 
a, B, y, by rotations of the directions of i,j, k, round the axis of a quaternion q-} = p. 
The expressions of a, 6, y, will be 
(54) a=piq, B=pq, y = pkq, 
where we suppose 
(55) pg = To" =A, 
As we have g= Kp, or p=Kq, we may apply formula (5) of § 1, which gives 
bDgt+tKap=0. 
Multiplying the first term by p, and the second by itsequal Kg, and consider- 
ing that by the known relation we have 
K(q) x K(dp) = K(dp.9), 
we get 
(56) peqt+K(<dg¢ 7=0 
Applying (5) to (<ip.g) we get 
>K(dp.g) + Kd (<dtp.¢g) = 0. 
