Sir W. Rowan Hamilton on Quaternions. 117 



diagonal of a parallelogram of which two adjacent sides have 

 any two given directions in the same given plane ; yet it is 

 desirable, for the reason mentioned at the beginning of the 

 last article, to know how to obtain the same general trans- 

 formation of the same symbolic relation, without having re- 

 course to geometrical considerations. 



Suppose then that any research has conducted to the rela- 

 tion, 



aa'u" -u>'ct'u = 0, (1.) 



which is not in this theory an identity, and which it is required 

 to transform. [We propose for convenience to commence 

 from time to time a new numbering of the formula, but shall 

 take care to avoid all danger of confusion of reference, by 

 naming, where it may be necessary, the article to which a 

 formula belongs; and when no such reference to an article is 

 made, the formula is to be understood to belong to the cur- 

 rent series of formulas, connected with the existing investiga- 

 tion.] By article 20, we may write the recent relation (1.) 

 under the form, 



S.*a'a"=0; (2.) 



and because generally, for any three vectors, we have the 

 formula (12.) of art. 22, if we make, in that formula, a." 

 = V./3/3', and observe that S(V./3|3'.a) = S(«V./3/3') = S.a/3/3', 

 we find, for any four vectors aa' /3/3', the equation : 



V(V.a a '.V./3/3')=aS.a'/3/3'- a 'S.a/3/3';. . . (3.) 



making then, in this last equation, /3 = a', fi' = u", we find, for 

 any three vectors, a «' a", the formula : 



V(V.«a'.V.a'a") = -a'S.a a 'a" (4s) 



If then the scalar of the product « «' «" be equal to zero, 

 that is, if the condition (2.) or (1.) of the present article be 

 satisfied, the product of the two vectors V.aal and V.ct'a." is a 

 scalar, and therefore the latter of these two vectors, or the 

 opposite vector V.u"x', is in general equal to the former vec- 

 tor V. aa', multiplied by some scalar coefficient b; we may 

 therefore write, under this condition (1.), the equation 



V.a''u' = bY.uu', (5.) 



that is, 



V.(a"-&a)a' = 0, (6.) 



so that the one vector factor a!' — bx of this last product must 

 be equal to the other vector factor a! multiplied by some new 

 scalar b'; and we may write the formula, 



a" = bu + b'x', (7.) 



as a transformation of (1.) or of (2.). We may also write, 



