429 



and as the perpendicular from the origin upon the tangent 

 plane is equal to 



\x + y + z ) i 

 we find the volume of the small pyramid, which has the origin 

 for its vertex and dS for its base, to be equal to 



x dydz 



But again, as 



3 



x 



dx = zd<f> + yd\, 

 dy = xd(j> + zdyj 

 dz = yd§ + xdfo 



the element dydz must be replaced by {x 2 - yz) dtyd-% under 

 the double sign of integration. Hence 



Thus it appears that the sector, generated by a radius vector 

 drawn from the origin, and having for its base the portion of 

 the surface bounded by the lines = 0, % = 0' = > X = X » 

 may be represented by the product ^ (p'x- 



" It has been observed that the functions obtained by di- 

 viding the two tresines by the cotresine possess properties 

 analogous to those of the trigonometric tangent. A remark- 

 able instance to this effect may be adduced in connexion with 

 the preceding theorem. 



" Let us put 



y z 



trit (0, x ) = - = t), and trit ( x , 0) = - = Z; 



x 



then 



d>] 



xdy - i/dx , 1V xdz - zdx 

 = — ?—f- — , and dZ, = 



, _ (x* - yz) d(f> - (y 2 - zx) dx xd<j> - zd% 



and 



