4-32 Sir W. Rowan Hamilton on Quaternions. 



77. To show the geometrical meaning of the equation (185.), 

 let us divide it on both sides by T.p^)}^ ; it then becomes, after 

 transposing, 



-SU.)39=SU.*3p.SU.pfl + S(VU.))p.VU.p9). (186.) 



Here, by the general principles of the geometrical interpreta- 

 tion of the symbols employed in this calculus (see the remarks 

 in the Philosophical Magazine for July 184-6), the symbol 

 SU . ijS is an expression for the cosine of the supplement of 

 the angle between the two arbitrary vectors >j and fl; and 

 therefore the symbol — SU . >3fl is an expression for the cosine 

 of that angle itself. In like manner, — SU .r^p and — SU .pfi 

 are expressions for the cosines of the respective inclinations of 

 those two vectors ij and 6 to a third arbitrary vector p ; and at 

 the same time VU.ijp and VU . p9 are vectors, of which the 

 lengths represent the sines of the same two inclinations last 

 mentioned, while they are directed towards the poles of the 

 two positive rotations corresponding; namely the rotations 

 from >j to p, and from p to 9, respectively. The vectors VU.jjp 

 and VU.p9 are therefore inclined to each other at an angle 

 which is the supplement of the dihedral or spherical angle, 

 subtended at the unit-vector Up, or at its extremity on the 

 unit-sphere, by the two other unit-vectors U>] and U5,or by the 

 arc between their extremities : so that the scalar part of their 

 product, in the formula (186.), represents the cosine of this sphe- 

 rical angle itself (and not of its supplement), multiplied into 

 the productofthe sines of the two sides or arcs upon the sphere, 

 between which that angle is included. If then we denote the 

 three sides of the spherical triangle, formed by the extremities 



of the three unit-vectors Urj, U9, Up, by the symbols, »)d, vjp, p9, 

 and the spherical angle opposite to the first of them by the 



symbol rjpfl, the equation (186.) will take the form 



cos >39= cos>)pcos pS-f sinijp sin pfl cos >]p9; . (187.) 



which obviously coincides with the well-known and funda- 

 mental formula of spherical trigonometry, and is brought for- 

 ward here merely as a verification of the consistency of the 

 results of this calculus, and as an example of their geometrical 

 interpretability. 



A more interesting example of the same kind is furnished 

 by the general formula (179.) iov four vectors, which, when 

 divided by the tensor of their product, becomes 



S(VU . a'V'.VU . u'u) = SU . a"'« . SU . a'a" 



-SU.«'V.SU.«"«; (188.) 



