500 Prof. Donkin on the Geomehical Interpretation 



and join QR. Then, by the same principles, we have C'TOriw 



Vm^ + n\q-\-li:= Vl^-\-m^ + n\q\ 



where g' represents the quadrantal rotation QR, If we sup- 

 pose 



, , , . t . , M-l ;':»«} >::*■;« roc-' 



then what we have just proved is that 



il + {jm + Jen) 



represents the quadrantal rotation QR, without alteration of 

 the length of the operand. 



Let QR cut YX' in T. Then in the right-angled triangle 

 QYT the preceding conditions give easily 



cosYQT=/, cosYQ=— 4=. 



whence, by Napier's rules, iiprfl Jbnis 



cosYTQ=w. 



Thus I and n are the cosines of the angles which the plane of 

 QR makes with the planes of z/s, xy\ whence it follows that 

 m is the cosine of the angle which it makes with the plane of 

 xz. In other words, /, m, n are the direction cosines of the 

 axis of the rotation QR; and an examination of the figure 

 constructed for any particular case, shows that they belong to 

 the positive axis of that rotation. ,. > 



We have thus interpreted the expression il+ (jm-\-/cn); and 

 the symmetry of the result shows that the interpretation of 

 {il+Jm)-[-/ai or o?{il + kn)+jm would have conducted us to 

 the same conclusion. Thus the associative principle of ad- 

 dition is established so far as quadrantal rotations are con- 

 cerned, and we may write il+jm + kn without brackets. 



Consider now the expression 



Hbir; 6dJ dv c°s ^+ «^" 9(/^-f> + ^n). 



The first term represents the operation of altering the length 

 of the operand in the ratio of cos fl to 1 ; and the second, the 

 compound operation of altering its length in the ratio of sin 5 

 to 1, and turning it through a right angle in a determinate 

 plane. Applying, therefore, the principles explained above 

 with reference to the interpretation of such expressions as 

 a-\-a'q'f we see that the complex symbol now under conside- 

 ration represents the operation of turning the operand, without 

 altering its length, through an angle fl, in a plane perpendi- 

 cular to the axis whose direction cosines are /, w, w. Let r 

 denote such an axis ; then we may write r • 



cos6+ siu6(«V+> + ^«) = (+,)%g^«53^56i^.,a 



