ii24 PROCEEDINGS OF THE AMERICAN ACADEMY 



A surface of the second order may then be represented by an equa- 

 tion of the form 



Sqcpq = c, 



where g) is a vector function of q of the first order. 



The most general form of such a function, qpp, will be a function, 

 homogeneous in q plus a constant vector y. But the homogeneous func- 

 tion is equivalent to a self-conjugate function q'^o, plus a term of the 

 form Vfo (Hamilton's Elements, § 349, (4) ; Tait, § 174). Now we 

 see that 



S(jgp(> = Se((jPoP + Vfo + 2y) = So(y„(» + 2 8;-^ = c, 



writing in the 2 merely for convenience. That is, the homogeneous 

 part of the function may be taken as self-conjugate. If we can next 

 transform the origin to such a point that y disappears, the surface will 

 be represented by the equation 



Sf^oP = ^' 



and in this case all the variable terms will contain q to the second 

 degree, so that satisfied by -|- p the equation will also be satisfied by 

 — q; i.e., the origin will be at the centre. To find this point, write 

 p -|- 5 for Q, and (dropi)ing the suffix of q^^n, but remembering that the 

 function is self-conjugate) 



SQ(f.Q -f 2 SQq)d -\-2SyQ-\- Sdcpd + 2 Syd = c. 



The terms 2 SoqrS and 2 S;'(i take the place of the old term 2 Sj'p; 

 and, in order that they may disappear, we must have 



Se((jp5 + 2') = 0; 



or, since q may have any direction, 



(p5 -|- 2' = 0. 



This is the condition that must be satisfied in order to transform the 

 origin to the centre ; and it gives in general a single finite solution for 

 5. I shall consider later the cases when this solution is indeterminate 

 or infinite, and the corresponding form of the surface. 

 Suppose now that the equation 



q,8-\-y = 



