154 Mr. Woodhouse’s Demonstration of a Theorem 
Let x, y, z be three rectangular co-ordinates, that determine 
the position of any particle of a solid body relatively to any 
three fixed axes, then the element of the solidity may be repre- 
sented by the paralellopiped zy x ; hence, s (solidity) = fff zy x, 
which triple integral must be taken first relatively to z, suppo- 
sing y and x constant; next, relatively toy, supposing x constant, 
having previously substituted in the function of z, for z, its va- 
lue in terms of x and y ; and, lastly, relatively to x, having 
previously substituted in the function ofy, fory, its value in terms 
of x. The order of these integrations may be varied at pleasure. 
In the present case, let the origin of the co-ordinates be the 
centre of the sphere; then s = fff zy x =ff (z f — z)yx\ and, 
if the integral be taken from the plane of x and y, where z — o, 
to the surface of the sphere, where z — z',s= ff z f y x; but the 
equation to the surface of the sphere is z* — s/ r z — x z — y 1 = o ; 
hence, s = ff*/ r ~ — x % — y z y x. 
Without the aid of some transformation, it would be extremely 
difficult to find the value of this double integral : the trans- 
formation is to be effected in the following manner. 
In the expression z'-xj, suppose first x = funct. (y, p) =-R ; 
then, regarding y as constant, x — J %' xy — zr ^ — j £ y ; 
and, when its integral is to be taken relatively to the value of 
x = R must be substituted for x in z r or v 7 r % — x z — y a . 
Again, lety = funct. =■ Q considering f constant,. 
j = 6 ; hence, z’ |?-J %y = z' ? ( 3 =-) Q; and, to inte- 
grate, substitute fory its value (Q) in z' The expression 
z' x y is thus transformed into another, relative to two variable 
