of Quaternions. 



skive axis is r; or, according to our previous nottition, 



il-\-jm-irkn = { + ^)i. 

 If the condition 



did not subsist, then, putting 



499 



we should have 



il-{-jm+kn={+ )^[x,f 



representing the compound operation of turning the operand 

 through a right angle in the plane perpendicular to r, and 

 altering its length in the ratio of ]x to 1, the direction cosines 

 of r being j^roportional to I, m^ n. 



The proof of this 

 is very simple, but 

 can hardly be given 

 without a diagram. 

 In the figure (of 

 which all the lines 

 represent arcs of 

 great circles), let X, 

 Y, Z be the positive X 

 poles of the rotations 

 i,j, k, or the points 

 in which the positive 

 axesof ,r, j/, z cut the 

 sphere; and let X', 

 7J be diametrically 

 opposite to X, Z. 



Now to interpret 

 mj + nk, take XZ' and XY to represent the rotations jt k; 

 then taking Q in YZ', so that 



sin YQ : sin QZ' : :m: n, 



and joining XQ, we have evidently, by the principles esta- 

 blished above, 



mj-\-nk— V tm^ ■^■n^.q^ 



where q represents the quadrantal rotation XQ. 

 interpret 



Next, to 



\^m^ + n\q + lit 



transport the rotation q in its own plane to QX', so that its 

 origin may be in the plane of the rotation /. Take QS a 

 quadrant, and join SX'. Take R in SX' so that 



sin SR : sin RX' : : \^m^ + n^:l, 



