OF ARTS AND SCIENCES. 2-13 



and would represent a pair of planes perpendicular to a.^, and hence 

 parallel to each other. 



For the sake of generality, I shall solve the equation for the centre, 

 and examine a couple of cases arising under it, by means of the cyclic 

 transformation. And although the result is far from satisfactory, yet 

 the solution contains some points of interest, and is worth inserting in 

 spite of its length. The well-known formula for this form of trans- 

 formation is (Ham. Elem., § 357 (5) and (8)) : 



qi^n = y^ -|- \Xq^ = (9 — SiX^)q -\- XSfiQ -{- [iSXq, 



where it is found that if we take c^<C2<C3, 



<^2 = — 9 + SV» 

 c^ = — g — TA/i, 

 c^= — g-\- Tin, 

 a^ = U(ATjM — juTA), 

 O2 =^ UV^.a, 



a, = U(lTjM + fiTl). 

 To solve 



9Q + ^^Qf^ = 7 



(changing for the sake of convenience the sign of y) by means of the 

 formulas of page 238. Let 



X ^^ X, fi = [I, and V = y. 



Here, again, qp is, of course, self-conjugate as before. Now we find 



cpX = gl~\- X% 



W = 9f^-\- V^ 



cpv = 9r + V^j'fi. 



And substituting these values : — . 



ScpXcpfiffv = S(^;. + X'^^i)(g^ -\- Xix^)(9y + XSyfi — ySX^ + fxSXy) 



= Sig'Xfi + 2 gXy + fi'X'){gy + ^Sj'^ - ySXfi + ^S^j') 



= S(/Vy + 2 g'Xyy - gXyy + fX'fiSyfi + 2 ffXySyfx 

 + Xy^yii — fXixySXfi — 2 gXyySXfji — fi^X^ySX^ -\- vec- 

 tor terms) 



= S(g^.Xixy — gX'^fi^.Xfiy — g^Xfiy.SXiA. -\- Xy.y.XySXfji). 



