230 BOOK OF MATHEMATICAL PROBLEMS. 



1095. Assuming the formulae for transforming from a system 

 of co-ordinate axes inclined at angles a, /?, y to another inclined 

 at angles a', /J 7 , y to be 



prove that 



1 = I* + I* + 1 + 2l a l a cos a + 2l a l } cos /3 + 21 f, cos y ; 

 with similar equations in m and n\ and that 



cos a = m^ + m a n a + m a n a 



+ ('>l a n 3 + >P>2) COS a + ( m 3 n i + m i n 3 } cos ft + ( m i n a + m a n i) COS 7 > 



with similar equations in n, 1; and I, m. 



1096. If ox*+by a + cz a become AX 3 + BY* + CZ* by any 

 transformation of co-ordinates, the positive and negative coefficients 

 will be in like number in the two expressions. 



1097. The equation 



ox*+ ... +Zofyz+ ... + 2a"x+... + d=0 

 will in general represent a paraboloid of revolution, if 



a b' c' b c a c a b' 



-, + - + r , = T, + - + - = - + r , + - = ; 

 a c b b a c c b a 



and a cylinder of revolution if, in addition to these conditions, 

 a" b" c" 



y+F*?- - 



1098. The surface whose equation, referred to axes inclined 

 at angles a, /?, y, is ax 8 + by 9 + cz* = 1, will be one of revolution if 



a cos a _ b cos ft c cos y 



cos a - cos /? cos y cos /3 cos y cos a cos y cos a cos ft ' 



1099. The surface whose equation, referred to axes inclined 

 at angles a, ft y, is ayz + bzx 4- c#y = 1 , will be one of revolu- 

 tion if 



a b c 



1 cos a 1 * cos /3 ~~ 1 cos y ' 



one, or three, of the ambiguities being taken negative. 



