430 HITCHCOCK. 



These equations complete the solution. Similarly, if (245) does 

 not vanish we can express tt in terms of (So, ^i, and /Ss. 



31. The remaining subcase supposes /3i an inflectional element of 

 the cones (3), so that g = 0. This is equivalent to saying that the 

 vector T of (171) shall be parallel to /3i. We then have, as the condi- 

 tion that /3i shall be a quadruple axis, by (181), 



S/3i7r^3 = 0, (194) 



for the scalar w cannot vanish if Fp is not reducible. (If so, we should 

 have X2 a factor of (171)). The condition (194) requires that tt shall 

 lie in the tangent plane .To = to the cones (3) at /Si. Since 183 is so 

 far any vector in that plane, we may take a/Sa = tt, and (171) becomes 



a^9XiX2 + U^lXs" + ^X2Xz + UX2^ • (195) 



Since we cannot now have any axis in the tangent plane distinct from 

 /3i, (for if so we should have four axes in a plane), we may assume ^2 

 an axis, and, by a properly chosen term in p, remove the term in X2^ 

 from the vector Fp. We then have 



a^3XiX2 + upiXs^ + ^3:2X3, (196) 



as the most general form for this sub-type. (The constants a and u 

 are, of course, altered by addition of a term in p). We have the 

 vector f and the scalars a and u at our disposal to determine two more 

 axes ^i and /Ss as in the former subcase. This, however, presents no 

 new difficulty. 



32. The methods already exemplified are amply sufficient to 

 impose on the two vectors (186) or (196) conditions that ^2 shall be a 

 double or a triple axis. As they are the most general possible vectors 

 of their types, as was shown, they contain all further special cases, 

 having /3i for a quadruple axis. 



As an example, let /Si = i, ^z = j, jSa = tt = k, and let a third single 

 axis be j + k, so that the three single axes are coplanar, and one of 

 them is in the tangent plane, (y = 0), to (3) at i. Let fif = 1 and </i= 0. 

 We find from (186) 



kixy + z^) -\- jy^ 



as the value of Fp, and the cones (3) become 



