130 SCIENCE PROGRESS 



To verify the last result, we have 



Kc~^c + K^^b = kU + Klb (by 3-21) ^-^K^rV + ^-KIv 



= ^K)o . K% + ^K% . Kir = kI^^o (by 3-24) 

 = KI'"^o = K^^o (scalars) = a. 



The assumption that K~^c = K^c is often useful, but it is allowable only 

 when we are finally reducing to scalars. But as the length c equals either 



~KrO or -K ^o, we also have 



a^-Kc c-Kb 'b-=--Kr o.Kr r--Kr o.Ky r = - Kr o = a; 



or a=~^C'o.B'r--C'r.B'o= -B'C'o (by 3-24 and 5-36) ; 



therefore (see 6-23) R^A = B'C. 



6'5. These equations are put in various ways as exercises ; and each 

 should be compared with the figure. The lines may be given other directions 

 than those shown. The symbols are freely commutative, but the zero-chords 

 are not the same as their scalar, zero. The reader may compare the per- 

 pendiculars from all the angles and should remember that 



[A,B, C, . . . ]S=IAS,BS.CS, . . . ]; e.g. [A . B . C .]S =^ AS.BS. CS. 



By putting 5 = o, or = r, we get the corresponding scalars. Observe the 

 method used for denoting a polybasic function. 



Extension to three dimensions and some important formula? are required 

 to complete this geometry. As regards both it has a rough resemblance to 

 quaternions ; but, nevertheless, it is really only ordinary trigonometry put 

 in another way. 



7'0. I have no space to pursue details and practical applications any 

 further, but must conclude with some necessary general notes. 



Trigonometry is dominated by two great Secondary Operations, namely, 



H = [o^]-! [o - I] [o^] = [o^]-^ [I + o]-^ [o^J ; (I) 



ifs[02]-i[i-0][02] 



= [o^]-^ [o -1] [I + o] [o-i] [I + o]-^[o-^]-^[o2]. (2) 



Both of them are of the CH~^ type ("conjugates," etc. in Group Theory), 

 of which ^ may be called the nucleus and f and f-^ the alae, and the general 

 property of which is that [f|C~^]** = C^^C""^- Thus, though apparently very 

 similar, H (hyperbola) and K (Circle) can now be seen to possess quite dis- 

 tinct molecular structures, the nucleus of H being merely the invert of the 

 First Primary Operation, ^ while that of K is the Third Primary — so that the 

 iteration of it gives periodic results. 



By attaching ro and its invert r-^O as additional alae we have 



Hr^ [ro] [H] [ro] -' = Vo^]£y^ 

 Kr = [ro] [K] [ro] -' ^ Vr^ - o\ (3) 



The iteration of H is obtained immediately, since [O — i]" = O — « ; 

 so that H^ = Vo^ - nr^. That of K^ was given in 3-0. The following 

 relation holds between H and K, namely 



K^^H'^K^H^. (4) 



1 The Primary Operations are derived successively from i + O, each by one 

 iteration (including inversion) and one change of base. The higher ones 

 appear to have been scarcely studied as yet. 



