226 DR. J. BOTTOMLEY ON THE EQUATIONS AND ON 



which is equivalent to 



dxdydz or to n z \ \ \ dx dy dz, 



that is n z x vol. of primitive solid between the corre- 

 sponding limits. 



The primitive solid and its z derivative being written in 

 the form 



z=<f>{x,y), 



z=n{l(x—a) +m(y — b) —en} -\-n z <f>(x } y). 



If we multiply the first equation by n z and subtract from 

 the second, we get the equation 



z(n z — i) + nlx + nmy— n(la + mb + nc) =o. 



Hence, if the primitive and its derivative have any points 

 in common they satisfy the equation to a plane. This 

 plane is at right angles to the primitive plane, for being 

 the angle between the planes, we shall have 



n nl z nm z 



cos &=—=== + ^ - nx /i— tf 



vi— n VI— n 



(l z + m z + n z -i), 



*/i—n z 



= o. 



The relation V^cos^V is only a particular case of a 

 more general theorem ; for let <j> (£, rf) be any arbitrary 

 function of f and 77, and let I z denote the integral 



<!>{£, V) d^d v d^; 



