OP ARTS AND SCIENCES. 241 



The vector of the ceutre has been found to be 



^ d q_ _^ 



~ "^ "i C2 "2 cg "3- 



But, if both d and Cj vanish, the centre must be at a determinate finite 

 distance in two directions, and at au indeterminate distance in the 

 other. The general equation 



Si'<jPo(? -\- 2 Sj'ji = c 

 becomes 



c^^^a.j) -\- CgS-Wgrt — 2 gSa/j — 2 kSa^Q = c. 



Reduce this to some point in the central line, by substituting 



e = e — -^ «2 — ^ «3. 



and our equation takes the form 



This represents a cylinder, because we may add xa^ to any vector 

 without affecting the equation ; as, indeed, we can see by inspection 

 of the unreduced form of equation for this case. We have thus found 

 again, by a totally different process, the same case of degeneracy con- 

 sidered on pages 235, 236, and 237. And in comparing our results we 

 must remember that the function there called cfn, and the one called qD^^ 

 here, are exactly the same ; at least, with the exception that the former 

 was multiplied by a constant, in order to make the constant term of 

 the equation equal to unity. 



If, in the equation for the cylinder, 



CgS^ajP -f~ ^s^'^'^sP — 2 g^(i.2Q — 2 h^a^Q = c,. 



C-Z Cg' 



the equation is a complete square, and equivalent to 



(v^e'Sa^e - ^) = ± (V^S«3P - ■^)' 



And this represents a pair of real planes, or a pair of imaginary planes 

 with the real intersection 



according as c^ and Cg have like or opposite signs. But these condi- 

 tions for the value of c are really the same that we found before, iu 

 VOL. xin. (n. 8. V.) 16 



