OF ARTS AND SCIENCES. 245 



To find 



= '^ fl -\-^ 9^^F7 — 3 (/y^^H + 3 m^Xy — <7}'S^f* 



— XSl^Sjiy -\- yS'^X^ — [xSlfiSXy. 

 "We have then 



mq) — ^y = tnd = m'y — m"(py -\- cp'^y = 3 g'^y — 2 gySl^ — 7^V^ — 

 3g'y-3 glSfiy + 4 yj'S V - 3 g^iSly + /j' + 2 ylS^y 



— 2 gySX^ -J- 2 g^SXy -\- ^X^Sfiy -{- Xfi^SXy. 



= — 7^V^ — y^Sfij' — gi^SXy -f- 5'V ~h f^X'^Sf.iy -\- Xfi'SXy 

 = X(fx'8Xy - gSfiy) + iu(rS^7 - ^rSl^) + y(g' - Xy). 

 And the complete solution for the centre is 



g __ \(/x^SAy - ffS^r) + M(A-SMy - gSA7) + X(g - T\^)(g + Txp.) 



From the values of Cj, c^, and Cg given on page 243, we see that, as we 

 found before under the rectangular form, d has a single finite value, 

 unless one of the roots Cj, c^, or Cg of qp^o vanishes. This cgclic solu- 

 tion for 5, because it contains no explicit rectangular vectors, is diffi- 

 cult to use ; and, indeed, often assumes a hopelessly indeterminate 

 form. 



"We can, however, obtain with sufficient ease in this cyclic form the 

 equations of the paraboloids. It was found under the rectangular 

 transformation that, in order that the equation of the second degree 

 should represent a paraboloid, we must have c^ or c^ disappear ; that is, 



g = SX|M or y = — TXfi. 

 Let us consider the latter case. "We know (Ham., § 357 (9), XXII.) 



8Xq{iq — Q^TXfj. = {(SXixoy + {SXo.Tfi + SfiQTXy} X (TV — 

 SXfi)- 1 = 2 SyQ + c, 



if 9=^ — Till" ; 



for gQ^ -\- SXqixq = 2 Syg -\- c 



is a form of the general equation of the second degree, and therefore 

 (SXfiQy + {SXQTfi + SfiQTXy = (TXix — SV)(2 SyQ -f c). 



