PEIRCE. — LINES OF CERTAIN PLANE VECTORS. 073 



v, the v curves are the possible lines of a lamellar vector the tensor of 



which is a function of v only. 



9k 9k 



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

 ly du 



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



the v curves are the lines of a set of solenoidal vectors the curls of which 



are expressible in terms of v only. 



Possible Ststems of Isothermal Straight Lines and 

 Isothermal Circles in a Plane. 



(1) Let ax + fty = 1> where a and ft are any functions of a single 

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

 may write 



9u _ -a 9u _ -ft a 2 + ft 2 



II,, 



9x a'x + ft'y' 9y~ a'x + ft'y' " '" (a' x + ft' yf 



V*(u) 2(aa' + (3/3') _a"x + ft"y 

 V a 2 + [3* a'x + (3'y ' 



(29) 



VHu) 

 If then — r-y- 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 only, and 

 we have a' = 0, or (3' = 0, or, in general, a" : a' = j3" : /?', so that 

 a = c j3 ■}- d, where c. and d are constants of integration. The equation 

 of the family of lines must be of the form (c{3 + d) x + fty = 1, and 



the lines all pass through the fixed point [ — , — — ) , which may be 



chosen at pleasure. If d =. 0, the lines are parallel. 



(2) Let x 2 -j- y 1 — 2 ax — 2 fty = y, where a, ft, y are functions of a 

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

 then we may write 



9u 2 (x — a) - 9u 2 (y — ft) 



9x 2a'x + 2ft'y + y> , 9y 2 a' x + 2 ft' y + y' ' 



h* = 



4 (a 2 + ft 2 + y) 



(2a'*+ 2ft>y+y'y 2 ' 



T 2 (w) _ 2aa' + 2 ft ft' + y' _ 1 2a" x + 2(3" y + y" 



h 2 a 2 + ft 2 + y a 2 + ft 2 + y' 2a'x+2ft'y+y'' 



vol. xxxvin. — 43 



(30) 



