494 Prof. Donkin on the Geometrical Interpretation 



part of the plane ; and the symbol q represents, indifferently, 

 rotation through an angle fl in atiy part of the plane. 



Now let q' = { + ,.>)^ represent in like manner the operation 

 of turning through an angle 6' = 2/37r in another plane whose 

 axis is r'. The two operations q, q' cannot in general be suc- 

 cessively performed upon the same operand line. In order 

 that this may be possible, it is necessary to place it at first in 

 the plane of the first rotation q, and in such a direction that 

 this first rotation shall bring it into the line of intersection of 

 the two planes, whereby the second rotation becomes possible. 



Let q" represent the single rotation which would have 

 brought the line from its initial position into that final position 

 in which it is placed by the successive rotations q, q'; we have 

 then the symbolic equation (or rather equivalence) g'q = q". 

 The successive rotations q, g' are (not identical with, but) 

 equivalent in effect to the single rotation q". If now a third 

 rotation q'" is to be performed, the rotation q" must (if neces- 

 sary) be transported in its own plane so as to make the com- 

 pound operation q"'q" possible, just as the first rotation q was 

 transported so as to make q'q possible. These conventions 

 enable us to assign a determinate rotation in a determinate 

 plane as the result (or product) of any number of given suc- 

 cessive rotations in given planes. But it is obvious that no 

 useful method of calculation could be based upon such assump- 

 tions, unless we could prove (as we can prove) that the asso- 

 ciative principle holds good with respect to the product of 

 rotations; that is, that {q"q')q = q"{q'q)- 



Let us now adopt a method of representing rotations (in 

 words or by actual diagrams) similar to that employed in the 

 previous paper. Conceive a sphere with arbitrary radius to 

 be described about the origin ; and, AB being any arc of a 

 great circle on this sphere, let " the rotation AB" signify that 

 rotation of a radius vector which would cause its intersection 

 with the sphere to describe the arc AB {JromAto B),oranj/ 

 equal and similarly described arc in the same great circle. [It 

 must be carefully borne in mintl, that we are now concerned 

 with the rotations of lines, and not, as in the former paper, of 

 solids] . 



Let ABC be any spherical triangle, and let the rotations 

 AB, BC, AC be represented as above by q, q', g*"; we have 

 then q'q=^q". It is easy to see that qq' does not represent 

 the same thing as q'q, and also to see what it does represent. 

 For producing AB to A' and CB to C, making BA' = AB, 

 BC' = CB, and joining C'A', we see that qq' represents the 

 successive rotations CB, BA', the effect of which is the same 

 as that of C'A'. If, then, we denote th s last rotation by q^^ 



