322 PROCEEDINGS OF THE AMERICAN ACADEMY 



S=(D CuTq' = <IJ CT.Q'VaQ. (5) 



4°. The direction of the normal to a surface of revolution is the 

 same as that of the perpendicular from the origin (which we still sup- 

 pose to be in the axis) dropped on the tangent to the meridian. If p' 

 be the tangent to the meridian, the projection of the radius vector q on 

 that tangent is 



— opo , and its length . 



p' '"'• ' ° Tp' 



Represent the perpendicular from the origin on the tangent by v. 

 Then if T»' be the length intercepted by the tangent, 



T-p' 

 T-pp' — S-pp' — V2pp' T^y pp' 



~ ry ■ ~ T-^p' ~ "ry"' 



and Tr = ^^. (6) 



Tp' ^ ' 



This expression gives the length of the perpendicular from the origin 

 on the tangent plane, and hence the element of volume swept by the 

 radius vector is given by 



\ Tf.wdgjTp'di = \ TV(>c.'.r/dg]d^ 



The finite volume will therefore be expressed by 



V=\<h Cu.TV(>(>'. (7) 



5°. Write q = q)(t, u) as the equation of a surface, with the con- 

 dition that t and u are two scalar indeterminates, the changes in t 

 determining a series of successive curves on the surface, these curves 

 intersected by another series of successive curves determined by the 

 changes in u. Then 



D,o = q\ 



will be the tangent to a curve of the first series, and 



D„(. = q', 



will be the tangent to a curve of the second series. The element of 

 area of the surface will therefore be TVQ^fj'^dicdt, and the finite area is 



S=ffTYe\Q'r (8) 



