66 VECTOR ANALYSIS. 



as will be evident on considering separately in the expression <. 

 the components perpendicular and parallel to 0, or on substituting in 



ii + cos q (jj + kk) + sin q (kj jk) 



for cos q and sin q their values in terms of tan ^q. 

 If we set, in either of these equations, 



we obtain, on reduction, the formula 



-h(2a& + 2c)/fc+(l-a 2 +6 2 -- 



in which the versor is expressed in terms of the rectangular com- 

 ponents of the vector semitangent of version. 



146. If a, ft, y are unit vectors, expressions of the form 



2aa-I, 2$8-I, 2yy-I, 

 are biquadrantal versors. A product like 



{2/3/3-!}.{2aa-I} 



is a versor of which the axis is perpendicular to a and ft, and the 

 amount of rotation twice that which would carry a to ft. It is 

 evident that any versor may be thus expressed, and that either a or ft 

 may be given any direction perpendicular to the axis of rotation. If 



<= {2/3/3-1}. {2aa-I}, and = {2yy-I}. {2/3/3-1}, 

 we have for the resultant of the successive rotations 



.<!= {2yy-I}.{2aa -I}. 



This may be applied to the composition of any two successive 

 rotations, ft being taken perpendicular to the two axes of rotation, 

 and affords the means of determining the resultant rotation by 

 construction on the surface of a sphere. It also furnishes a simple 

 method of finding the relations of the vector semitangents of version 

 for the versors <, , and ^ r .$. Let 



Then, since 



/> _a 

 "i ~~ 



^ ' 

 cc.p 



which is moreover geometrically evident. In like manner, 



