120 Sir W. Rowan Hamilton on Quaternions. 



present theory, we ma}' introduce the following considera- 

 tions respecting conical surfaces in general. 



Whatever four vectors may be denoted by a, a!, /3, /3', we 

 have 



V(V.a«'.V./3|3') + V(V./3/3'.V.aa') = 0; . (1.) 

 substituting then for the first of these two opposite vector 

 functions the expression (3.) of art. 24, and for ihe second 

 the expression formed from this by interchanging each a with 

 the corresponding /3, we find, for any four vectors, 



aS.a'/3/3'-a'S.a/3/3' + /3S./3W-/5'S./3a a ' = 0. . (2.) 



Again, it follows easily from principles and results already 

 stated, that the scalar of the product of three vectors changes 

 sign when any two of those three factors change places among 

 themselves, so that 



S.ufiy = — S.«7/3 = S.ya/3 \ . * 



= -S.yj8« = S./3ya= -S.j8ay.J * ' ' K * 

 Assuming therefore any three vectors i, x, X, of which the 

 scalar of the product does not vanish, we may express any 

 fourth vector a in terms of these three vectors, and of the 

 scalars of the three products axA, i a X, ix«, by the formula : 



a S. ixX = iS.ax\ + xS. i«X + aS.ix«. . . (4.) 

 Let a be supposed to be a vector function of one scalar vari- 

 able t, which supposition may be expressed by writing the 

 equation 



« = <K0 ; ( 5 -) 



and make for abridgement 



the forms of these three scalar functions^ /^/g depending on 

 the form of the vector function <$>, and on the three assumed 

 vectors » x X, and being connected with these and with each 

 other by the relation 



♦ (o = »/i w + */«« + y&.c* • •. • • ( 7 -) 



Conceive t to be eliminated between the expressions for the 

 two ratios of the three scalar functions f^f^f® and an equa- 

 tion of the form 



FC/iC). fM / 8 (0) = o (8.) 



to be thus obtained, in which the function F is scalar (or real), 

 and homogeneous; it will then be evident that while the 

 equation (5.) may be regarded as the equation of a curve in 

 space (equivalent to a system of three real equations between 

 the three co-ordinates of a curve of double curvature andan 



