PEIRCE. 



ON CERTAIN CLASSES OF VECTORS. 



303 



1 — ex 



=■ parameter. 



All the Hues of such a system pass through a fixed pouit (1/c, 0) ou 

 the X axis. 

 Again : let 



^2 -I- y2_ 2aa:- 2/?y-y = (42) 



where a, (B, y are functions of a single parameter u, represent a family 

 of circles in the xy plane so that 



2 (x — a) 



3u 



9x 2a'x + 2j3'y -h y' 



Hy-(3) 



9u 



9y 2 a' x + 2 /i' y -\- y' 



.,^ 4(a2 + ;8^ + y) 

 '" (2a'x + 2/3'y + y)-^ 



8 a' (aj — a) 



5'm 2 _^_^_ 



9x^ ~ 2 a'x + 2 yS' 2/ + y' ~ (2 a> x + 2 /3' y + y'f 



4 (a; - af (2 g^' a: + 2 /3^\y + y") 

 {2a<X+2lS'y + y'f 



L(u) _ 2aa' + 2/3(3' + y' 2 a" X + 2 /B" y -\- y" 



K' ~ a2 + y8== + y 2a'a: + 2/3'y + y' 



(y-^)(2a^x+2^-y + yO 

 4y(a2 + y82 + ^) 



(43) 



The first term in the second number of (43) is already expressed as a 

 function of u : the sum of the last two terms is a function of u if, and 

 only if, a' = 0, ft = 0, so that a is a constant (c) and (42) takes the 

 form 



(x-ay- + y' = u. (44) 



This represents a set of concentric circles with centre at same jioint 

 on the X axis. 



The Equation A„ = y. 



If the tensor of a solenoidal vector, symmetrical about the x axis, is to 

 have at every point of each of its lines a value constant for that line, the 

 tensor is a function of the parameter of the lines and the equation of the 

 family must be found among the solutions of equation (24). 



