494; Letter from Sir W. R. Hamilton to R. T. Graves, Esq. 



the product-line coincides with each of them ; and its am- 

 plitude is the sum of theirs ; because the spherical triangle 

 degenerates into a single arc, two angles vanishing and the 

 third becoming equal to two right angles. If the amplitudes 

 of the factors are also supplementary to each other, the am- 

 plitude of the product is tt, and its sine vanishes; the product 

 has therefore in this case no lengthy or radius, but becomes a 

 jnire real, and at the same time a negative quantity ; because 

 fj]' sin p" = 0, /jJ'cosf)" = — [J'. The factors are in this case 

 of the forms 



{jju cos p, fji, sin p cos i^, fx. sin p sin <p cos ^, /x. sin p sin <p sin \I/), 

 and 



( — /*' cos p, [xJ sin p cos ^, ^' sin p sin <$» cos J/, ju,' sin p sin <^ sin 4/) ; 

 ami their product is 



(-^/x,',0,0, 0)=-/Xia'. 



Making the factors equal, ?'=— , />(''=/!*, vve find that 



(0, fx. cos <f), jiA sin 1$ cos rf/, jw> sin ($ sin \I/)2 = — /x^. 

 The square of a pure imaginary is real and negative : the 

 square root of a real negative is a pure imaginary, having a 

 determined length, but a wholly undetermined direction. The 

 square root of — 1 is any one of the infinitely many pure ima- 

 ginaries, of the form 



s/ — \=i cos <^ +y sin ^ cos ■^ ■\- h sin ^ sin 4/. 

 In general, however, the square root of a quaternion has 

 only two values, which differ only in sign: for if 



(«",6",c", ^") = (a,Z',c,fZ)2, 

 then 



a" = a2 _ ^,2 _ p2 _ ^ 2^ j;/ ^ 2 a Z> c" = 2 « c rf " = 2 « ^/ ; 

 a/^2 ^ im ^ ^/2 ^ ^n ^ ^2 ^ ^,2 ^ ^2 ^ ^2^ 



and n^ cannot vanish, except in the case of the pure negative 

 square, b" = c" =: d" = 0, a" < 0. 



Multiplication will be easy if we are familiar with the rules 

 for the product of two pure imaginaries. This product is, 

 by (B.), 



{OAc.d) {0,b',c',d') = {-bb'-cc'- dd',cd' - dc\dV - bd\ be' - cU) ; 

 the product-line is perpendicular to the plane of the factors; 

 its length is the product of their lengths multiplied by the sine 

 of the angle between them : and the real part of the product, 

 with its sign changed, is the same product of the lengths of 

 the factors multiplied by the cosine of their inclination. 



Finally, we may always decompose the latter problem into 



