246 



PROCEEDINGS OF THE AMERICAN ACADEMY 



It is evident from the forms of the terms of the first member that the 

 resolved parts of 7 in the i-ectanguhir directions Wlfx and U(lT/t -\- 

 f/Tl) can be eliminated by taking a new origin ; and the y reduced to 

 the third direction U(ATjm — juTl). 



Now 



Fig. 3. 



T(;.T^ — ,tT;.) = TVT(m — v>^) 



= 2 TXfi sin ^ <^ = V^2 'PV(1 — cos <^), 



= \J2 {T')4i -{- Tin SV)*- 

 If k be the length of 7, our equation becomes. 



(SXhq)- + (s;.t)T,t -f S/<ot;.)2 

 = \/2k{T}.fi — sv)^ T-i ?.,/(s;.oT,i — S/;c'T;. 4- c) 

 = V2"i'T-u^(Tv — s;.,i)Hs(;.«T/t — ,<c'T;.) + 4. 



This is an elliptic paraboloid ; for, if cut by a plane perpendicular to 

 V/lju, or U(/'.T^t -f- ^Tl), the section is a parabola. But, if by a plane 

 perpendicular to U(^Tj« — /iTP.), it is an ellipse. 

 To obtain the hyperbolic paraboloid, let 



The general cyclic form of the equation of the second degree, in this 

 case, reduces to . 



By a transfer of the origin in the plane of l and /<, 7 may be reduced 



to a direction perpendicular to that plane ; and our equation to the 



form 



S^pSjup = nSlfjiQ -\- c. 



