to Non-Metrical Vector Algebra. 129 



according as JS falls on Oa or Oct' '. We may, therefore, 

 consider the latter equation as the definition of cosine. 

 Similarly, let us define the sine of by the tensor of the 

 normal bJS r , writing 



Bin^=8in(a,b)==|*JV|, .... (11) 



a, b being points on the unit conic k, as before. (The sine 

 and the cosine of an angle Wm constructed anywhere can 

 at once be reduced to the above by joining the T-points of 

 O'l, O'm with O, and so on.) Using these familiar symbols, 

 however, we must not yet confound them with the familiar 

 trigonometrical functions, until we have learnt that they are 

 identical with them. Such, in fact, they will be shown to 

 be presently; and their analytical identity will become 

 complete after an appropriate measure has been introduced 

 for ft still to be defined as a magnitude. 



At any rate we know already that if ft, ft, 6 r stand for 

 the zero angle, the straight, and the right angle respectively, 

 we have 



cos #o=l> cos ft =—1, cos ft ==0, . . (12a) 



although we do not know, for example, what the numerical 

 value of ft. is. Again, since ObN is a right-angled triangle 

 and Ob is a unit vector, we have, by the generalized 

 Pythagorean theorem, for any ft 



sin 2 0+ cos 2 = 1 (12b) 



Moreover, again in virtue of the distributive property, we 

 can deduce for our sin and cos the familiar additivity 



Fis\ 5 a. 



»/ 



theorems. For this purpose it is enough to declare that if 

 a = bOi and (3 = aOi be co-vertical angles having the side Oi 

 in common, aOb is the svm of these angles ; and similarly 

 for the difference. In fact, if /, a, b (fig. 5 a) be the end- 

 Phil. Mag. S. 6. Vol. 38. No. 223. July 1919. K 



