and its application to the Geometry of three Dimetisions. 125 



If we consider the two right lines which represent the quan- 

 tities X -\- s{y) -\- n{z) and 5(a? + s(^) + ;?(«)), we shall see that 

 either of them is obtained from the other by causing it to turn 

 round the axis OA through an angle of 180°. The sum of 

 the two, therefore, compounds a line in the direction of that 

 axis ; and their difference is a right line in the plane perpen- 

 dicular to OA. Thus it appears that the product (1+5(1)) 

 {x-{-s{y)->tn{z)) denotes a right line, which, as well as the fac- 

 tor 1 + 5(1), coincides with the axis of rotation; whilst the 

 line denoted by the product {l—s{\)){x + s{y) + n{z)) lies, like 

 the factor 1—5(1), in the plane perpendicular to that axis. 

 Here we have a geometrical interpretation of the strange- 

 looking results, 



(l+5(l))(^ + 5(.t/) + «(-2r)) = (l+5(l))(.r+j,) 



{\--s{l)){x+s{y)) = {\-^s{\)){x-y) 



10. Lest it should be supposed, however, that the explana- 

 tions just given of these seemingly anomalous results have arisen 

 accidentally out of the geometrical interpretation assigned to 

 the symbol 5, I proceed to show that these results admit of 

 being explained by reference to the analytical conditions im- 

 posed upon 5 at the outset. I do so, because in the course of 

 the inquiry we shall be led to notice some general principles 

 useful in other systems of algebra as well as in the one be- 

 fore us. 



On reviewing the operation which conducted to the result 



{x + s{y)){a^ + s[y'))=a^' + s{f), 



we observe that this equation holds good, whatever distributive 

 operation 5 may be, provided it satisfies the equation 5^(a) = a. 

 Now this equation has two purely algebraic solutions 5= + 1 

 and 5= —1. It follows, then, that we may write successively 

 + 1 and — 1 instead of 5 in it, and so we obtain at once the 

 two modular equations (1.). And further, whatever equation 

 we find subsisting, in which s appears along with real quanti- 

 ties, it must continue to hold good for 5= + I and 5= — 1. 

 Thus, for instance, from the equation 



c*W = Hyp. cos x-\-s Hyp. sin x 



we derive both 



fr'=Hyp. cosa:+ Hyp. sin a: 

 and 



e~*= Hyp. cos x~ Hyp. sin x. 



Suppose now that we had before us some expression of the 

 form (1 +5(l))!p(*, a-, 3/, z) ; we might expect to find in all trans- 

 formations of it the operator (1+5(1)) still remaining, or ca- 



