PEIRCE. — LINES OP CERTAIN PLANE VECTORS, G73 



r, the V curves are the possible lines of a lamellar vector the tensor of 



which is a function of v only. 



9h 9h 



If -—^ =: and -^ = 0, the u curves are the lines of a set of sole- 



noidal vectors the curls of which are expressible in terms of u only ; and 

 the V curves are the lines of a set of soleuoidal vectors the curls of which 

 are expressible in terms of v only. 



Possible SrsxEMS of Isothermal Straight Lines and 

 Isothermal Circles in a Plane. 



(1) Let ax + jiy = 1, where a and fi are any functions of a single 

 parameter u, represent a family of straight lines in the xy plane, then we 

 may write 



9u -a 9il —13 ,2 a^ + /3^ 



9x a'x + /3' «/ ' 9y a' X + Id'y' " (a' X + (3' y)' 



V%it) 2 (a a' + 13 (3') a" X + (3" y 



hj a' + (3' a>x + /3'y' 



(29) 



V%ti) . 



I 



If then — jp^ is to be a function of u only, the last term in the second 



member of this last equation must be expressible in terms of u onl_y, and 

 we have a' = 0, or /3' = 0, or, in general, a" : a' = yS" : /?', so that 

 a = c /3 -\- d, where c and d are constants of integration. The equation 

 of the family of lines must be of the form (^ /S + d) x -\- fty ^= 1, and 



the lines all pass through the fixed point ( - , -\ , which may be 



chosen at pleasure. If (/ = 0, the lines are pai allel. 



(2) Let x'^ -{- y'^ — 2 ax — 2 (3 y — y, where a, f3, y are functions of a 

 single parameter u, rejjresent a family of circumferences in the xy plane, 

 then we may write 



9u 2(x — a) 9u 2(y — /3) 



9x 2a'x + 2p'y + y'' 9y 2a'x+ 2 /3'y -\- y' 



4 (a^ + iS'^ + y) 



(2a'a:4- 2/i'y + y)" 



V^(«) _ 2aa' +2I3[3' + y' 1 ^ 2a" X -\- 2(3" y + y" 



h^ ~ a^ + yS^ + y «■-= + /i2 + y ' 2a'a;+ 2/i'y + /' 



vol,, xxxviti. — 43 



(30) 



