OF ARTS AND SCIENCES. 335 



This result, substituted in formula (8), gives 



S =JJa^ V[T^«a'(//x)2 + T'Vaa'(f,'xyi (51) 



y X 



Let us now write a system of equations for the quarlric surfaces. 

 For this purpose we have only to substitute, in equation (49), such 

 functions of x and of y as siiall cause o, when y varies, and q, when y 

 is constant and x varies, to describe conic sections ; and any functions 

 which identically satisfy the conditions (fiX)' rt (f^^)^ = 1? {/.'FY 

 ± {f^x)' = 1, or any equation of the second degree in f^x, {f.^-fi'y)', 

 and {f./'-f^y), will evidently serve our purpose. We may write then, 

 for a system of quadric surfaces, the following equations: — 



I. (>:=:« cos x -\- a sin Xy 



cr = p cos y -\- y m\ y; the ellipsoid ; 

 II. Q = aChx -\- (xShx, 



o = ^ cos y -\- y sin y ; the parted hyperboloid, having a for its 

 principal axis ; 

 III. Q = aShx -|- crCha:, 



(7 = ^ cos y -\- y sin y; the unparted hyperboloid, having a for 

 its principal axis ; 



IV. Q = a- — \- ox, 



<T = j5' cos y -{- y sin y; the elliptic paraboloid, having a for its 

 axis ; 



V. Q = a— -j- GX, 



cr = (iChy -|- j'Sh^; the hyperbolic paraboloid, having a for 

 its axis ; 

 and finally I write 



VI. Q = ax -\- (j"^, 



(T = j3 cos y -\- y sin y, as being analogous to IV. and V., though 

 not representing a quadric. 



The function TVo'^n'^ derived from I., — if we observe that Voff' = 

 |3j', and make the substitutions f^x = cos x, fix = — sin x, etc., in 

 (50), — now becomes 



TVq\q'2 = sin X W{ah' sin^ x -\- W-c^ cos^ x) 

 = s sm cc V (a'' cos'' a:), 



