iiMii\uiAy,(3^'i%v\ ii).of Quaternions. 'lialnoCI .'toi 501 

 where arari aw ^Bta^tiitq 9fflB8 9i!i yd >n9dT .Hp nio[ bna 



The quaternion iaorn.iiijp ^ril 8la889iqdi *tp eiarfv/ 



ls}-\-ix+ji/ \-kz "^^''O 



is always reducible to the form 



^ i? ow jfid?/ nailj; 

 /*(^cos 9+ sin 9(?7+j7W + ^n)J, 



where , ^ 



by the assumptions 



fx,= Vw^+w^ + i/^ + z^i r= \^w'^ + i/'^-\-z% ftsind=r, /t*cos9=w, 



/ __ »i _ 72 _ 1 ^ 



x"^ y z r' 



and therefore it represents a rotation, combined with an alte- 

 ration of length in the ratio of jw, to 1. 



It must be observed, however, that we have so far only in- 

 terpreted the expression : / " "' ''V ' 

 '- * ^ cam aaiflni Hgt 

 w + {ix +ji/ -\- iz), 



and that we are not at liberty to remove the brackets without 

 first establishing the associative principle of addition with re- 

 spect to symbols o( alieration q/'lenglh, such as w, and sym- 

 bols of alteration of length combined with quadrantal rotation^ 

 such as ix^ &c. But this is very easily done, and as easily for 

 any rotations as for quadrantal. >'^ 



Let ABC be any spherical triangle, and let q, q' represent 

 the rotations AB, AC. Let a, b, c be numerical quantities. 

 Then the equations '-* uojJit) 



(a + bq) + c^' = a + [bq + eq^) = (« + c^) + bq, 



are easily seen to express, that if a parallelepiped be con- 

 structed with three edges coinciding in direction with the radii 

 drawn to the points A, B, C, and proportional in length re- 

 spectively to a, &, Cjthen the diagonal of the parallelopiped may 

 be obtained indifferently by taking the sum of any one of these 

 edges and the diagonal of the parallelogram constructed on 

 the other two. 



We have now established* all that is necessary to give de- 

 monstrative force to geometrical conclusions deduced from 

 algebraical calculation with quaternions. This paper has, 



* The geometrical proof of the distributive ^voi^evty^ expressed by such 

 equations as , ' ; 



i{a-\-jb)-=:.ia-\-l<bt ...>... >i* 



Sec, is so easy and obvious^that I leave it to the reader. 



