JUO The Rev. K. Carmichael on the Quadrature of Surfaces 



ordinary works upon the differential and integral calculus. In 

 the elaborate treatise upon this subject by M. lVAbbe Moigno 

 (Paris, 1844, vol. ii. p. 235), the expression is investigated, by 

 the usual analytical method, transformation of coordinates, from 

 the well-known expression in rectangular coordinates, 



da=\/{l+p* + g' i )dxdy, 



and is given in the following shape, 



( ]a= \/{r°~sin*6+ sin 2 0(D fl r) 2 + (D^f}r d6 d4>. 



A short geometrical deduction of this expression, whose merit 

 I have great pleasure in sharing with my friend, Alexander Jack, 

 Esq., A.B., may not be unacceptable to the student. 



Let P be any point on the surface. Through the axis OA 

 and OP describe a plane, and 

 round the axis describe, with 

 the same line, a cone. The 

 surface may then be supposed 

 to be divided into its elements 

 by planes and cones consecu- 

 tive to these respectively (the o~~ ~~a 

 planes all passing through the axis and the cones round it), half 

 of one such element being represented by iPt'. Then, remem- 

 bering that the planes cut the cones orthogonally, we have 



da = Yt, . Vc' . sin tVi'-Vc . P*' . V(l- cos 2 riY), 

 whence 



da=?i. IV. \/(l - sin 2 1V0 . sin 2 i'Po') = ./(Pi 2 . Pi' 2 -Oi 2 .oV 2 ), 



o and o' being the points where the sphere described round the 

 origin with radius OP intersects the consecutive radii vectores 

 to the points t, t! ; or 



d a = \/ [{r 2 sin 2 0# 2 + ( V) 2 <ty 2 } {rW 2 + (D e r) V6> 2 }-(D fl r) W (Dtffdtf], 

 or finally, 



da= */{r 2 sin 2 0+ sin 2 6{Yf e rf + (D^) 9 }rd0 dcf). 



3. From this expression we may readily derive that for the 

 perpendicular from the origin upon the tangent plane, in polar 

 coordinates. In rectangular coordinates it is known to be 

 p _ s—px-gy 



V^l+Z + Sr 2 ) ' 

 but the transformation of this to polar coordinates would be 

 troublesome and tedious. We may easily derive the required 

 expression from the volume of the elementary cone, for 

 Vd<r=.r*m\dd6d4>, 



