56, 57] 



RESULTANT OF INFINITESIMAL ROTATIONS. 



203 



In the spherical triangle ABC we have 



. 0> a . <B 



sm ~ sm 



Fig. 45. 



sin COB sin CO A sin AOB 



The preceding results are much simplified if the rotations are 

 infinitely small. 



We shall first prove that two equal infinitely small rotations in 

 the same sense about axes infinitely near each other may be regarded 

 as equal. Suppose the axes 

 first parallel, and perpendic- 

 ular to the paper which they 

 cut in A and B. Let a point P 

 be rotated about A through 

 the angle do to P', and 

 through the same angle about 

 B to P". The arcs PP' 

 and PP" differ by the amount drdo } if dr is the difference between 

 AP and BP. They are inclined to each other at an infinitely small 

 angle BPA } and as the sides PP f and PP" are infinitely small, and 

 differ by an infinitely small quantity of the second order, P'P" is of 

 the second order. If the axes are inclined to each other at an 

 infinitesimal angle, there is a third component 

 perpendicular to P'P", which is likewise of the 

 second order. Therefore the theorem is proved. 

 The theorem of rotations about intersecting 

 axes may then be stated. Two infinitesimal 

 rotations about intersecting axes are equivalent 

 to a rotation about an axis in their plane, 

 the order of rotations being immaterial. To 

 find the position of the axis of the resultant 

 rotation, we have, by 5) 



dco, d(o a dco 



sin COB sin CO A sin AOB 



If we lay off on the axes OA and OB (Fig. 46) 

 and OQ proportional to the rotations do and do 2 , the 

 tions show that OC is in the direction of the diagonal of 

 gram constructed on OP and OQ as sides and the resultant 

 is proportional to the diagonal OE. 



Therefore the resultant of two infinitesimal rotations 

 intersect is found by the parallelogram construction, or 

 of addition of vectors. This process may be. extended to 

 of infinitesimal rotations whose axes intersect. 



lengths OP 

 above equa- 

 a parallelo- 

 rotation do 



whose axes 

 by the law 

 any number 



