I B, thet t ane. 5 
5 a al 
13 
sealar or real number. Let 3 be the revolving line, consi- 
_ dered in its original position ; 3’ the same line, after it has 
revolved through the angle a round the axis a. The part, or 
component, of 3, which is in the direction of this axis, is that 
which was denoted lately by (3,; and the formula (a), when 
multiplied by — a, gives, as an expression for this part, 
B: = 3 (8 — aBa), (g) 
because it has been supposed that a? = —1. This part of 6 
remains unaltered by the rotation. The other part, or com- 
ponent of (3, is, in like manner, by (b), 
B2 = 4(8 + aBa) ; (h) 
and this part is to be multiplied by cosa, in order to find the 
part of 3’, which is perpendicular to a, but in the plane of a 
and 8. Again, multiplying by a, we cause (3, to turn through 
aright angle in the positive direction round a, and obtain, for 
the result of this rotation, 
a. = 3 (a8 — Ba) ; 
an expression which is the half of that marked (b), and which 
is to be multiplied by sin a, in order to arrive at the remaining 
part of the sought line 6’, namely, the part which is perpen- 
dicular to the plane of a and £. Collecting, therefore, the 
three parts, or terms, which have been thus separately ob- 
tained, we find, 
PB’ = Bi + (cosa + asina) B, 
= 3 (B—aa) + 3 cosa (B + aBa) 
3 sin a (aB—Pa) 
= (cos$).8— (sii ae aBa + cos 5 sin 5: (aB—Ba) ; 
that is, 
= (cos $ +a sin 5) B (cos 5 —asin > ; (i)* 
* [Note added during printing.|—The printing of this abstract having been 
delayed, the Author desires to be permitted to append the following remarks: 
