502 On the Geometrical Interpretation of Quaternions. 



however, already extended to such a length, that I will only 

 give one simple instance of such reasoning. 



Let ABC be any spherical triangle, and q, q\ q" represent 

 the rotations AB, BC, AC. Then we have, as before ex- 

 plained, q" = q'q. But if we put AB = 6, BC = 9', and call 

 /, OT, w, /', m'i n' the direction cosines of the positive poles of 

 AB, BC, then we have seen that 



q = cos 9 -f sin fl ( // +jm + kn) , 



and q' is similarly composed of accented letters ; hence 



q'q=: cos Q cos 6'— sin sin &'{ll' + mm' + nn') 



+ i{ls'mQcos6' + l' s'm$' cos Q + {m'n— mn') sin 9 sin 9'} 



+j{m sin Q cos $' + m' sin 6' cos 9 + {n'l—nl') sin 9 sin 6'} 



-{■k{n sin S cos 6' + n' sin 6' cos 6 -\-{lin' — I'm) sin 9 sin 9'}. 



This is the quaternion symbol of the rotation AC ; its first 

 term therefore expresses cos AC, which agrees with the fun- 

 damental theorem of spherical trigonometry ; and the terms 

 affected by z, J, k are proportional to, and determine, the di- 

 rection cosines of the positive axis of AC. The product qq' 

 in like manner represents the rotation C'A', if A', C' be points 

 taken in AB, CB produced, so that BA' = AB, BC'=CB. 



The reader who is familiar with Sir W. Hamilton's re- 

 searches on quaternions will observe that these and similar 

 results, which are here primmy, appear in his system of inter- 

 pretation as secondary ov polar ; and the converse would easily 

 be shown to be true also. (See particularly Irish Transac- 

 tions, vol. xxi. part 2, p. 80-86.) The reason of the differ- 

 ence is apparent. In Sir W. Hamilton's geo?netrical system, 

 ifj, k are not symbols of operation, but represent concrete 

 units, namely, unit lines in fixed directions,. so that ix -\-jy -\- kz 

 represents also, not an operation, but a concrete quantity, 

 namely the radius vector of the point whose coordinates, re- 

 ferred to axes coinciding with the directions of /, j, k, are 

 X, y, 2. The quaternion w + ?.r +73/ + ^2; therefore represents 

 what may be called a couple^ or sum of two heterogeneous quan- 

 titieSi namely, an abstract length w, and a directed line 

 ix+jy + kz. It is therefore not capable of a geometrical in- 

 terpretation. This circumstance, however, as Sir W. Ha- 

 milton has abundantly shown, does not at all interfere with 

 the use of quaternions in obtaining geometrical results. 



In the system of interpretation which has been explained in 

 this paper, every symbol is regarded as representing an ope- 

 ration ; and every term of the quaternion, as well as the whole 

 expression, has a geometrical interpretation. In fact, a qua- 



