13 



scalar or real number. Let |3 be the revolving line, consi- 

 dered in its original position ; j3' 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 /3i ; and the formula (a), when 

 multiplied by — J a, gives, as an expression for this part, 



i3i=:l(/3-«/3a), (g) 



because it has been supposed that a^ = — 1. This part of ^ 

 remains unaltered by the rotation. The other part, or com- 

 ponent of /3, is, in like manner, by (b), 



j3, = ^(i3 + ai3a); (h) 



and this part is to be multiplied by cos a, in order to find the 

 part of |3', which is perpendicular to a, but in the plane of a 

 and j3. Again, multiplying by a, we cause /3.2 to turn through 

 a right angle in the positive direction round a, and obtain, for 

 the result of this rotation, 



ajSa = 1 (aj3 - jSa) ; 

 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 /3', namely, the part which is perpen- 

 dicular to the plane of a and /3. Collecting, therefore, the 

 three parts, or terms, which have been thus separately ob- 

 tained, we find, 



j3' = j3i + (cos a + a sina) j32 

 = I (iS-ajSa) + 1 cos a + a/3a) 

 + Isina (aj3 — j3a) 



= ^(cos 1) . /3 - \%m I j . a/3a + cos | sin | . (a/3 - /3a) ; 



that is, 



/3' = ^cos I + a sin Ij /3 ^cos | — a sin |^ ; (i)* 



* \_Note added during printing.'] — The printing of this abstract having been 

 delayed, the Author desires to be permitted to append the following remarks : 



