in the usual process for forming a Plane Surface. 109 



values of/?, two corresponding values of q, and two permissible 

 values of E and of ft ; and the solutions will be 



A n +p'.B n =W.(q>r +(i ', 



An+p".Bn=W.(q»)" +(i "; 



from which A n and B n may be found. 



Forming the equation for/?, we find 2p 2 +p + 1 = 0; the roots 

 of which may be expressed in the form 



*=-j±~- {#$-*=*> Ah 



f=-^{VT+v=i.AY- 



3 1 



and substituting in the formula -g- + -^p, 



Let 



cos a = ^1, sin a = v/ 1, (a = 69° 1 7' 43" nearly) ; 



then it will be found that 



~~ vH = cos 3o1 ' ~~^k = sin 3a ; 



and the expressions become 



/>'=— sj-- . (cos a — V — l.sina), 



q'= —\/-3 • (cos 3a + V — 1 . sin 3a), 



y=— v/-i-.(cosa +\/^T.sina), 



3 "= -yl . (cos 3a- i/^T. sin 3a). 



It appears here that (p') 3 = q", and (p 1l ) 3 —q'; and this is verified 

 in the following manner. From the equation for p just employed, 



0=jp3 + i^+l^ 



The sum is 

 Therefore (p') 3 = j — ^-p'. But, in the second term of the 



