334 PROCEEDINGS OF THE AMERICAN ACADEMY 



The volumes enclosed by the surfaces of the hyperboloids will be 

 considered later in connection with the discussion of the general ec^ua- 

 tions of the quadric surfaces. 



To obtain the volumes bounded by the general ellipsoids and elliptic 

 hyperboloids, it is only necessary, in the results of 22^* and 23'*, by a 

 well-known pi-inciple, to change a^ into ac, or b- into be, where c is the 

 length of the third semi-diameter. 



24"^. The general discussion of the areas and volumes of a very 

 important class of surfaces will be facilitated by writing, in each case, 

 a quaternion equation of the surface in terms of two independent sea 

 lar variables. 



Let «, ^, y — three vectors diverging from the origin — be any 

 three axes of the surface whose generating vector is q. Let a = the 

 vector — complanar with q and a — whose extremity describes the 

 section of the surface in the plane of j3v. "Write the ec^uation of this 

 section in the form 



^ = ^fzV + rf^h 



and the equation of the surface in the form 



Q — «/i^ + fl/"-^' (49) 



whereyp/^, etc., are separately functions of a single variable. In this 

 equation (49), the two scalar variables are evidently independent of 

 each other, a new value of either determining a new point on the sur- 

 face ; Q will describe, if x remain constant, a section parallel to the 

 plane of ^)', if y remain constant, a section whose plane contains a. 

 For convenience we may suppose « _L {i _L 7. We shall now have 



Da-p =z n\ = tangent to meridian at extremity of (), 



Dj,p = ()', = tangent to parallel of latitude at extremity of q ; and, 

 performing the differentiation, 



q\ = ((fi'x + Gf/x, p'2 = a'fy:, 

 whence 



Yo\'J, = yaa'f,xf,'x + Yaa'f.xf.'x, 



and, since Yaa" and Yaa' are perpendicular to each other, 



VV/2 = (/2^)'[V^«c/(//x)2 + VW(/,':r)^], 

 or 



TYq\q', z=f^x V[T'aa'(f,'xy + T'Yaa'(f,'xyi (50) 



