OF ARTS AND SCIENCES. 321 



II. Applications of Quaternion Analysis to Eectification 

 OF Curves, Quadratuke of Surfaces, and Cubature op 

 Solids. 



1°. If p = (jp (t) be the equation of any curve in space, it is easily- 

 shown that the complete derivative of p, relatively to the scalar variable 

 involved, is the tangent to the curve ; — 



D,o = (/ = (f' (t). (1) 



Here Tq' is the derivative, and To'dt the element, of the arc of the 

 curve. Hence, any length of arc will be 



s„ = JT(,'. (2) 



2°. Again, the element of double area swept by the radius vector 

 will be TVQQ'dt, and any finite area swept by q will be 



= h fTVQo'. (3) 



If the surface be plane, we may change the origin of vectors to a 

 point in the plane, and so find an area measured from the new origin 

 and limited by the limiting positions of the new generating vector. 



Let d be the vector of the new origin (from any origin whatever in 

 space). Then the generating vector will have the form 



tir = (? — 8, 



and the finite area will be 



t t 



A=h fTVnrvr' = \ CtV(q — d)Q'. (4) 



to <o 



3°. If a plane curve be revolved about a given axis lying in Its 

 plane, the surface generated will be one of revolution. If p = gi(;) 

 be the vector equation of the curve, referred to a point in the axis of 

 revolution, To'd^ will be the element of arc of the meridian (or gener- 

 ating curve). Let ic be tlie distance of a point of the curve from the 

 axis of revolution, and g) the angle of revolution. Then itdcf will be 

 the element of the parallel of latitude, and tcdrpTo'dt the element of 

 area. Hence, if a is a unit-vector parallel to the axis, the area of a 

 portion of a zone will be 



VOL. XIII. (n. s. v.) 21 



