164 Professor Baker, On the reduction of homography 



and also 



r= x'+ Fz'+ Vt', ^1 =^ xi + Fz:, + 7«i, 



rj'= y'+ Gz'+ Vt', r]^=iJi + Gz^ + Vt^; 

 then tiie equations of the transformation ^/u.~^ take the forms 

 f'= li, 7]'= 7^1, z = am-'^ii + hm-q^ + Cz-^ + Df^, 



assuming that the roots a, ^ of the equation in a, 



{C - a) {D'- a) - CD = 0, 



are different from one another, and from unity, we can take p, p' 

 and q, q' so that 



pC + p'C _ pD + p'D ' _ qC + q'C _ qD + q'D' 



p p q q ^ 



and thence, by 



Z'= pz'+ p't', Zi = pz-^ + p% 



T'= qz'+ q't', Ti = qz^ + q't„ 



the last two equations of ^/x~^ are replaced by 



Z'= Aij_ + 57^1 + aZ^, T'= A'i^ + B'rj^ + ^T^, 

 where 



A = {pa + p'a') m,-^, B = {pb + p'b') m, 



A'= {qa + q'a') m-^, B'= {qb + q'b') m, 



so that, finally, if we put 



A Ti 4 Ti 



A' Ti' A' Ti' 



the equations of ^/x-^ are 



which are the equations of an axis-range perspective in which every 

 point of the line ^ = 0, r = is unchanged, and every plane 

 through the line ^ = 0, 77 =- is unchanged. The latter Hne is the 

 axis given in the notation first used by the parameter — 6^, be- 

 longing to the quadric Sq ; the former line, we easily see, is given by 



am-'^x'+ bmy'+ (c - 1) z'+ dt'= 0, 



a'm-^x'+ b'my' + c'z' + {d'- 1) t'= 0. 



The transformation (Xj) = fx (x), equivalent to 



a?i = mx, y^ = m-^y, z^ = z, t^ = t, 



