138 Mr. H. Cunynghame on a 



where g=(fi 1 n/'R ) 2 , and E and L have the former meanings. 



When only the total current is under investigation, the 

 method followed by Lord Kayleigh (Phil. Mag. May 1886) 

 possesses advantages. I find it difficult, however, to under- 

 stand how the increased resistance can become of serious 

 moment. For, above a certain speed, the current amplitude 

 is increased ; whilst, below that speed, its reduction, from that 

 given by the linear theory, appears to be, in copper wires, 

 quite insignificant in general. 



The remainder of this paper I must postpone, it being 

 already of a reasonable length. 



XV. On a new Hyperbolagraph. By H. Cunynghame*. 



IT is a not unfrequent want to be able to find the rectangle 

 of greatest or least area contained between a curve and 

 any given rectangular coordinate axes. In several problems 

 connected with motion and pressure in steam-engines this is 

 very useful ; and even in political economy the graphic repre- 

 sentations of monopoly-curves depend on the maxima and 

 minima of this nature.. 



For the solutions of such problems it is often very useful to 

 be able to describe any rectangular hyperbola, the axes being 

 given. To effect this, I have constructed a machine which is 

 capable of drawing these curves with considerable accuracy. 

 It depends on a mathematical property of the rectangular 

 hyperbola, which, so far as I am aware, is new. From a fixed 

 point O let a line P be drawn to meet a fixed line A B in P. 

 Take P Q perpendicular to A B, and make P + P Q = a 

 constant length. Then Q is the locus of a rectangular hyper- 

 bola whose asymptotes are equally inclined to A B, and such 

 that if the said asymptotes be taken as axes, ay = 2(OM) 2 . 

 This is easy to prove ; for if PM = ^, PQ = z/, and 0Q = 6 and 

 OM = A; then 



y = a-OY, 



= a—\/ / a 2 -\-tt' 2 , 

 or 



c? — 2ay +y 2 = a 2 -+ x 2 , 



y 2 -2ay =a?, 



which is the equation to a rectangular hyperbola. 



* Communicated by the Physical Society : read June 12, 1886. 



