PEIRCE. — ON CERTAIN CLASSES OP VECTORS. 303 



1 — ex 



= parameter. 



y 



All the lines of such a system pass through a fixed point (1/e, 0) on 

 the x axis. 

 Again : let 



x 2 + f-2ax- 2fty-y = (42) 



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

 of circles in the xy plane so that 



9u _ 2 (x — a) 9u _ 2 (y - ft) 



9x~ = 2a'x+ 2ft'y + y 1 ' 9y~ = 2a'x + 2 ft' y + y 1 ' 



4(q 2 + /3 2 + y) 



" - (2a>x + 2(3>y + yiy' 

 9 2 U 2 8 a' (x - a) 



9x 2 2a'x +2ft'y + y' (2a'x + 2ft'y + y') 2 



A(x- a) 2 (2 a" a: + 2 ft" y + y") 



(2a>X+2ft'y + y'f 



L(u) _ 2aa' + 2 j3 ft' + y' 2 a" x + 2 ft" y + y" 

 K a 2 + /3 2 + y 2a'x+2ft'y + y> + 



(y-ft)(2a'x + 2ft'y + y>) 

 Ay (a 2 + ft 2 + y) 



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 _ a ) 2 + y 3 = w. (44) 



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

 on the x axis. 



The Equation h u = 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). 



