4S0 Sir W. Rowan Hamilton on Quaternions. 



And instead of the square of the tensor of the quaternion 

 prj—Qp, we may write any one of several general expressions 

 for that square, which will easily suggest themselves to those 

 who have studied the transformations (already printed in this 

 Magazine), of the earlier and in some respects simpler equa- 

 tion of the ellipsoid, proposed by the present writer, namely 

 the equation 



T(<^ + ^x) = x2_,9. eq. (9.), art. 38. 



For instance, we may employ any of the following general 

 equalities, which all flow with little difficulty from the princi- 

 ples of the present calculus: 



-{pri-6p)(r)p-pQ) = (rip-p&){pri-$p) 

 = {r}^ + Q^)p^-py}pQ-6prip 



= {n + Q)Y-{rip+pri){dp+pQ) 



= (>j' + flV-2S.r]^flp 



z={yi + Q)y-4^S.rip.SJp 



= (l-fl)y + 4S(V.,,p.V.p5); .... (171.) 



and which all hold good, independently of any relation between 

 the three vectors v}, $, p. 



76. As bearing on the last of these transformations it seems 

 not useless to remark, that a general formula published in the 

 Philosophical Magazine of August 1846, for any three vectors 

 a, a', «", namely the formula 



aS.«'a"-a'S.a"« = V(V.aa'.a"), eq. (12.) of art 22, 

 which is found to be extensively useful, and indeed of constant 

 recurrence in the applications of the calculus of quaternions, 

 may be proved symbolically in the following way, which is 

 shorter than that employed in the 23rd article: 



V(V.a«'.a") = |(V.a«'.a"-a"V.«a') = l(««'.a"-«".aa') 



= !«(«'«" + «V)-i(aa" + «"«)«' = aS.a'a"-a'S.«"«. (172.) 

 The formula may be also written thus: 



V.a"V.a'a = «S.a'a"-«'S.««"; . . . (l73.) 

 whence easily flows this other general and useful transforma- 

 tion, for the vector part of the product of any three vectors, 



a, a', a": 



V.a"a'a = aS.a'a"-«'S.a"« + «"S.a«'. . (174.) 



Operating on this by S.a'", we find, for the scalar part of the 

 product of any four vectors, the expression : 



S .«"'«"«'« = S'. «'"«. S . u'x" - S.«"'a'. S.«"« + S. a"'«".S.««'.( 1 75.) 



