Sir W. Rowan Hamilton on Quaternions. 121 



auxiliary variable t, which latter variable may represent the 

 time, in a motion along this curve), the equation of the cone 

 which passes through this arbitrary curve, and has its vertex 

 at the origin of vectors, is 



F (S. a x A, S. « « A, S. I x it) = 0. . . . . (9.) 

 Such being a form in this theory for the equation of an arbi- 

 trary conical surface, we may write, in particular, as a defini- 

 tion of the cone of the nth degree, the equation: 



2(A / , >ff>r (S.axX)P.(S.i«A)«.(S.ix«) r )="0; . (10.) 



p, q, r being any three whole numbers, positive or null, of 

 which the sum is n; A Ptq<r being a scalar function of these 

 three numbers; and the summation indicated by % extending 

 to all their systems of values consistent with the last-mentioned 

 conditions, which may be written thus: — 



sin^? it — sin q n = sin r ir = ; "1 



P = ®> <7 = 0, r>0; K . . . (11.) 

 p + q -+- v = n. J 



When n — 2, these conditions can be satisfied only by six 

 systems of values of p, q, r ; therefore, in this case, there enter 

 only six coefficients A into the equation (10.); consequently 

 five scalar ratios of these six coefficients are sufficient to par- 

 ticularize a cone of the second degree ; and these can in ge- 

 neral be found, by ordinary elimination between five equations 

 of the first degree, when five particular vectors are given, such 

 as a', a", a'", a lv , a v , through which the cone is to pass, or 

 which its surface must contain upon it. Hence, as indeed is 

 known from other considerations, it is in general a determined 

 problem to find the particular cone of the second degree which 

 contains on its surface five given straight lines: and the ge- 

 neral solution of this problem is contained in the equation of 

 homoconicism, assigned in the preceding article. The proof 

 there given that the six vectors « . . a y are homoconic, when 

 they satisfy that equation, does not involve any property of 

 conic sections, nor even any property of the circle : on the 

 contrary, that equation having once been established, by the 

 proof just now referred to, might be used as the basis of a 

 complete theory of conic sections, and of cones of the second 

 degree. 



27. To justify this assertion, without at present attempting 

 to effect the actual development of such a theory, it may be suf- 

 ficient to deduce from the equation of homoconicism assigned 

 in article 25, that great and fertile property of the circle, or of 

 the cone with circular base, which was discovered by the ge- 

 nius of Pascal. And this deduction is easy; for the three 



