212 The Rev. R. Carruichael on the Quadrature of Surfaces 

 or, 7 being the angle made by the radius vector with the axis of z, 



sin 7 . d<x, 



ff 



a minimum, is given by the partial differential equation 



qjr — 2pqs +p' 2 t = 0, 

 whose integral is known to be 



•»,(•) +yF,(r)=il, 

 representing the gauche surface generated by a right line which, 

 gliding upon two fixed directrices, remains constantly parallel to 

 the plane of the axes of x and y ; as indeed might be anticipated 

 from a consideration of the question in its second form. 



In the same manner it might be shown that the surface which, 

 within given limits, renders the double integral 



ff 



a minimum, is given by the equation 



</>F 1 (r) + 6'F 2 (r) = l. 



If it be proposed to investigate the property of this surface 

 corresponding to the character of the generation of the analogous 

 surface in rectangular coordinates, as the latter character is ex- 

 hibited by the supposition z = const., so the former property may 

 be investigated by the supposition r = const. Let, then, the sur- 

 face be supposed to intersect a sphere described round the origin, 

 and let the nature of the curve of intersection be examined. If 

 we resolve any element into its rectangular components, one 

 such component is r dd, and the other r sin 6 dcf). Let i be the 

 inclination of the element to the meridional plane described 

 through its extremity and the fixed axis, and it is evident that 



tcmi -_ rsinfl# _ r g (c) 



c being the radius of the sphere; or the tangent of the angle of 

 inclination of the curve to the meridional plane is proportional to 

 the sine of the angle made by the radius vector with the axis. 



7. It may be well here to indicate certain desiderata, the 

 knowledge of which might lead to the discovery of some inter- 

 esting properties of surfaces. 



The measure of curvature at any point of a surface is expressed 

 in rectangular coordinates by the formula 



1 _ rt-s* 

 we have no corresponding expression in polar coordinates. Such 



