228 BOOK OF MATHEMATICAL PROBLEMS. 



may separate into (real or impossible) linear factors : which is the 

 case when h = A, B, or C. 



But if two of the three coincide as B = C' t then when 7t = 7> 

 the two factors become coincident, or either expression is a complete 

 square. The conditions that this may be simultaneously the 

 case in the former expression give us 



(B -a)a' = -tie', ifec., 



b'c , c'a a'b' .>!/,% n ., 



or B = a -- 7- = jr = c -- T if a , 6 , c be all finite. 



O> C 



If a' = 0, then b'c must also vanish; suppose then a', &' = 0, 

 therefore B = c, and we must have 



(c -a)x*+(c- b) y* - 2c'xy 

 a perfect square, whence 



c fs = (c-a)(c-b). 



In the case of oblique axes, inclined at angles a, /?, y, we 

 must have 



h (x 2 + y*+z*+ 2^cos a + 2c cos /? + 2;r?/ cos y)-ax 9 - ... - 2ayz- . . . 

 a complete square. 



It follows that the three equations 



(h - a) (h cos a - a'} = (h cos J3 - b') (h cos y - c'), 

 (h b) (k cos ft b') = (h cos y c) (h cos a a'), 

 (h - c) (h cos y - c') = (7i cos a - a') (A cos /5 - 5'), 



must be simultaneously true, and the two necessary conditions 

 may be found by eliminating h. 



1089. If there be two systems of rectangular co-ordinates, and 

 #i> 0, 3 be tne angles made by the axes of x', y', z with that of 

 2, and <j, < fl , < 3 the angles made by the planes of zx' t zy', zz' with 

 that of zx\ then will 



with two similar equations. 



