340 HITCHCOCK. 



Fs contains a component along (3i or /.-, and terms free from z. Simi- 

 larly it may be shown that Fi and F2 may be expressed according to 

 theorem 3. 



Theorem 4. Conversely, if a quadratic vector be expressed in 

 rectangular coordinates, and if its components X and Y are free from z, 

 it has a central axis. 



Proof. The two-dimensional quadratic vector iX -\- jY has three 

 axes. Call them 71, 70. and 73. Take i along 71 (by a rotation of 

 axes in the xy plane if necessary, which cannot alter the fact that X 

 and Y are free from Z). We may then write 



iX + jY = iiax-' + bxy) + j{a'x'' + b'xy + c'V) 



It is evident from the form of the term pS8p that we may now, by a 

 proper choice of 8, remove the component along i, namely by taking 

 8 = ai + bj. The given quadratic vector will then have the form 

 jY -\- kZ, hence possesses three axes in the plane perpendicular to 71. 

 Similarly we may show that the given quadratic vector has three axes 

 in the plane perpendicular to 72 and similarly for 73. But k is the 

 common line of intersection of these three planes, and is itself an axis, 

 since A" and Y vanish when x and y are both zero. That is, k is a 

 central axis. 



Theorem 5. The vectors 5i, 82, 8s, which convert, respectively, 

 Fi into F2, F2 into ^^3 and F3 into Fi, are axes of the two-dimensional 

 vector iX + j Y. 



Proof. Inspection of the three 5's obtained in proving theorem 2 

 shows that they are perpendicular, respectively, to F3, Fi, and F2. 

 But in the proof of theorem 4 it appeared that the three 7's have the 

 same property. Hence the directions of the 6's coincide with the 

 respective 7's. 



Theorem 6. If a quadratic vector Vippdp + pS8p has a central axis, 

 and if a proper choice of rectangular coordinates be made, a value 

 can be found for 8 (which does not alter the axes), such that the 



differential equation -^ = — takes the Riccati form. 



dx X 



Proof. As in the proof of theorem 4 we may obtain the quadratic 

 vector in the form 



i{ax~ + bxy) + jia'x' + b'xy + c'y^) + kZ 

 By taking 8 = bj and adding the term pS8p this becomes 



