a Surface of the Second Order into itself. 211 



tions become 



MM'a,'2 = 7'S.r J + a«'//j — a'Bz^ — «7' w, 



MM%= 13/3' x^ + 78V I -/SS'-i -/3'7zt', 



MI\I'^2 = /37'.r , + 7«Vi - /3«'~ 1 - 77'M'i 



values which give identically ^gj/g — 2'2W2=*'iJ'i~'^i^^i' More- 

 over, by forming the value of the determinant, it is easy to verify 

 that the transformation is in fact an improper one. We have 

 thus obtained the equations for the improper transformation of 

 the surface ocy—zw = into itself. By writing arj + iy^ ^i~il/i 

 for ^1, i/i, &c., we have the following system of equations, in which 

 (a, b, c, d), («', y, c\ d') represent, as before, the coordinates of 

 the poles of the plane sections, and ]\P = «^ -f- i^ + c^ + d^, 

 M'*=fl'2+i'2 + c'2 + rf'2, viz. the system* 



M.M\=(aa' — bb' — cc'—dd')x\ + {ab' + a'b + cd' — c'd)7/i 



+ {a(^ + a'c + db'—d'b)si + {ad' + a'd + bc'—b'c)iv^ 

 MMVa = {aV + a'b —cd' + cld)xi + {— ad + bV —cd— dd')yi 



+ {bc' + b'c—da' + d'a)zi + {bd' + b'd—ac' + a'c)w 

 MM'^,= {ad + a'c-db' + d'b)!i-i + (bc' + b'c-ad' + a'd)y^ 



+ { — aa' — bb' + cd — dd')z^ + [cd' + dd— ba' + b'a)wi 

 MM'w^= {ad' + a'd— be' -h b'c)^^ + {bd' + b'd— cd + c'a)y, 



■•\r{cd' + c'd—ab'-^db)z^-\-{ — ad—bb' — cc' + dd')w^, 



values which of course satisfy identically x^ + y^ + z^ + w^ 

 ^ss.x^-\-yy-\-z^-\-w^, and which belong to an improper trans- 

 formation. We have thus obtained the improper transformation 

 of the surface of the second order x'^-\-y'^-\-z'^-\-uf'-=.Q into itself. 

 Returning for a moment to the equations which belong to the 

 surface xy — 2i<; = 0, it is easy to see that we may without loss of 

 generality write a = /8 = a' = /3' = 0; the equations take then the 

 very simple form 



MM'x2 = 7'Sa'i, MM'y2 = 78Vp ^l^z^=-^iw^, 

 WWw^-=-Wz„ 



where MM'= V^— ^8"/— y'8'; and it thus becomes veiy easy 

 to verify the geometrical interpretation of tlie formulse. 



It is necessary to remark, that, whenever the coordinates of 



* The system is vcrj' similar in form to, but is essentially different from, 

 that which could be obtained from the theory of quaternions by writing 



MWiWj+ix2+jyi+kZi)= 

 {d+ia+jb+kc){w + ix+jy+kz){d'+ia'+jb'+kc'); 

 the laHt-mentioned transformation is, in fact, proper, and not improper. 



