to Son-Metrical Vector Algebru. 131 



-where S is the cross of the b-line with the conic tangent 

 at a, the upper or the lower sign being used according as S 

 falls on the segment aT n or outside it (i. e. below a or 

 beyond T n ). It is not difficult to see that these two de- 

 finitions are equivalent. In fact, if n be normal to a, and 

 therefore co-terminal with aS, we have 



S = a -|- \n = /jib = /^(a cos 6 + n sin 6), 



'•whence /* cos = 1, fi sin = \, .'. X= sin 0/cos 6, and 



a£=S — a = Xn, 



so 



that 



i rv . sin 

 — i ' cos 

 proving the statement. 



Notice that the jump of tan from oo to — oo when # passes through 

 a right angle (0 r ) obtains the following elegant representation : When b 

 (fig. 6) approaches n, the point S approaches T n , the value of tan 6 thus 

 increasing to +oo , and when b exceeds n, the point S passes across the 

 J'-line thus indicating the jump of tan 6 to — oo . It will be remembered 

 from the construction of the projective scale (P.V.A.) that, on every 

 straight, to one side of T corresponds oo , and to the other side — oo . The 

 discussion of the obvious geometrical correlate of the fact that cos 6, for 

 instance, does not exceed the values ±1 may be left to the reader. It 

 will be enough for this purpose to note that the pencil of all lines T n b 

 (leading to our previous N) is limited by the two tangents T n a, T n a' '. 



Let us now pass to compare angles with one another, so 

 -as to be able to treat as an algebraical magnitude. And to 

 begin with let us give a definition of angle equality. This 

 can, in the first place, be based upon ab=cos# itself. 



Limiting ourselves to concave angles we may define two 

 angles a and ft as equal if their cosines are equal, no matter 

 where their vertices are situated *. In symbols, if 



ab=lm, (13) 



we will say that angle a, b = angle 1, m. 



The simplest sub-case of such a relation occurs when 1 

 and a, m and b are equal to one another but situated in 

 different places. This gives, as a corollary of the above 



* Notice that every angle having its vertex on the !F-line (being of 

 the form 1, 1) is a zero angle, unless one of its sides is the TMine, when 

 the angle is a right angle. Of. supra. 



K 2 



