SURFACES OF THE SECOND DEGREE. 



87 



and we obtain 



[^J-X^(r-,F-.Z')[^J=X.(^-/^;). 



Substitute from the last of (39) for [jfl, and develope <f>{X', V, Z'), 



removing the term which contains it to the left hand side; which 

 gives 



ca= 



r, + F, jaX" + br" + cZ'' + &c. &c.) 



■(41), 



answering to (8) in my former paper. 



We shall afterwards proceed to some applications of these general 

 formulas, and now enquire into the several varieties of the equation (3), 

 and the criteria for distinguishing between them. The following table, 

 immediately to be explained, gives a synoptical view of the various 

 caseSi inhcrfHt i->9.R :i;f<>1 '"\'r^hr' 5r .-^.v: 



