502 



PROCEEDINGS OF THE AMERICAN ACADEMY. 



Hence 



(/r.O[(I-A)-(I-A')] 



= ((/r.OA)-(I-A') — ((/r.OA'*)-(I-A*) — c/r-OA'-AI. 



Hence iinally 



20^ = -OA-(I-A') -OA'*-(I-A*) -<0>A'-AI 



-<0>A'*-(I-A*) — OA-(I-A') + OA-A'I. 



If the expression <C> • ^ is desired, care must be exercised to insert the 

 dot between the first two vectors of each triad. Hence ^^ 



20>.^ = 2 «>-A)-A' + 2 «>. A'*).A* - OA'-A + OA-A', 



O.^ = (O-A).A' + (0'A'*)-A* + §(OA-A' - OA'-A). (158) 



Some Projective Geometry, and Trigonometry. 



64. We may discuss very briefly the relations between our non- 

 Euclidean measure of angle and the projective measure as determined 



by logarithms of cross-ratios. Let 

 us consider Figure 30 first as a 

 Euclidean and second as a non- 

 Euclidean diagram. The two fixed 

 lines a, /3 are drawn so that they 

 are perpendicular from the Eu- 

 clidean point of view. The initial 

 line from which angles are meas- 

 ured is taken as the bisector of 

 one of the right angles; this line 

 and its perpendicular through the 

 origin will be taken as axes of x 

 and y. The pseudo-circle appears 

 as a rectangular hyperbola with 

 the equation x~ — y- — 1 . The angle between the initial line and any 

 radius in the pseudo-circle in Euclidean measure will be called 6, and 

 tan 6 = y/x. Now in non-Euclidean measure, if this angle be called 

 (f), we have seen that tanh = y/x. Hence we have the relation 



tan d = tanh (f). 



^^ The form <0>A«(I«A') may be written as a sum of triads of the type 

 aA«(ef) or a(A«e)f. Now by (35), a*(A'e) = — (a'A)«e. Hence the in- 

 sertion of a dot in 0A'(I*A') gives — (•0*A)« (I«A') or — (<()>• A) -A'. Iff 

 the form ^A'A'I, the dot goes between ■Q> and I, since A-A' is a scalar. 

 But as I is the idemfactor, we have simply <3>A«A' as the result. 



Figure 30. 



