OF ARTS AND SCIENCES. 223 



and if we assume any arbitrary values for x and y, we shall have a 

 scalar equation to determine the corresponding value of z. Our equa- 

 tion, then, represents a surface for the same reason that any one equa- 

 tion between Cartesian co-ordinates represents a surface. It is almost 

 needless to add that, since q.Q may be any vector function of q, the 

 converse proposition is true : that any surface may be expressed by an 

 equation of the form 



SQ(fQ ■=■ constant. 



The degree of the surface is higher by unity than the degree of the 

 function qjQ, — understanding, as usual, by the degree of a surface the 

 greatest number of times it can be cut by a right line. For suppose 

 that (jpp is of the n"* degree, and we want the intersections of the sur- 

 face with the line, 



Q =z xa. 



(pQ may be divided into the sum of vector functions, each of which 

 is homogeneous with regard to q, the highest being of the n"* degree. 

 We shall thus have 



SgqjQ = Sgcp'Q -\- Snq)"Q -j- &c. 



= TqTcp'q cos <p,^ + TQTq)"Q cos <^^,^ + &c. = c. 



Now we may substitute xa for q in this equation. But cos < '^ 

 depends only on the direction of q and cp'Q, and since we may write 



cp'Q = cp'{TqUq) = T\>(p'{\Jq), 



the direction of q)'Q depends only on that of q. Hence, the cosines in 

 the above equation are independent of x. 



Now Tcpo contains x to the same degree in each term as it does q ; 

 that is, to the n"" degree in the highest term. Thus the equation 



TqTcp'q cos <^P^ + TQTrp"Q cos <^,^ + &c. = c 



is an algebraic equation of the degree {n -\~ 1), which gives (n -|- 1) 

 solutions for x, or (»i -[- 1) distances at which the surface is cut by 

 the line 



Q = xa. 



