116 Mr. W. Spottiswoode on the Geometrical 



and X, Y, Z will be the coordinates of the extremity of the 

 diagonal of the parallelopiped formed on the sums of the com- 

 ponent coordinates as its edges. From these two facts it ap- 

 pears that straight lines lying in the same straight line are to 

 be added as in ordinary algebraical geometry, while the sum 

 of any other set of straight lines inclined to one another at any 

 angles is the closing side of the polygon formed by placing 

 the beginning of each line at the termination of its prede- 

 cessor. In fact, lines are to be added as forces are equilibrated 

 in statics. In accordance with this principle, the sum 



ix -\-jy + kz 



will represent the diagonal of the parallelopiped described on 

 the line ix y jy, kz as its edges ; and since moreover 



{ix +jy + kzf= ~ (* 2 +/ + s 2 ) = - r\ 



therefore also 



k+j^+**='(—-)*r, (5.) 



which, according to the principles of the present calculus, re- 

 presents not merely a line in a plane perpendicular to r, but 

 a line which has been brought into its position by means of a 



rotation through an angle = - in that plane ; or, in other 



words, about an axis whose direction-cosines are x : r, y : /', 

 z:r; and finally, the sum 



iso -f- ix +jy -f kz 



will represent the diagonal of the square whose sides are 



w + ix -\-jy -f kz 



(these two lines being obviously perpendicular). The length 

 of the whole line is consequently 



(w 2 + # 2 +3/ 2 + * 2 #=p, (6.) 



and its direction makes an angle, whose tangent is =r:w, 

 with the direction of w; the whole quaternion will therefore 

 represent a line whose length is p, which has been turned 

 through an angle = tan _1 (r: w) in the plane, the direction- 

 cosines of whose normal are x : r, y: r, z: r. The expres- 

 sion (1.) may also be written as follows : 



{cos 0-f sin Q(il +jm + /m) }p, .... (7.) 

 where 



6= tan" 1 ^:^) ~) 



7 r • • • - ( 8 - 



x : l—y. m = z : ?i = r J 



