93 



CONJUOATE Functions and Canonical Transformations. 



By David A. Rothrock. 



(Abstract.) 



It is known that any function, ^ (Z), of a complex variable, Z = x -|- 

 i J, may be separated into a real part ch (x, y i an imaginary part, i<!>2 'x, y ), 



and that 9,, 9, each satisfy Laplace's equation '- — ? + — -^ = o* A very 



f5x2 (5y2 

 elegant geometric interpretation of these two functions (p^, <p2 may be had 



by equating each to a third variable C : 0i (x, y) =: C, (J2 (x, y) = C. Each 



equation then represents a surface for any point of which Laplace's equation 



is true. By developing C ^= ^j (x, y) into a power series in the vicinity of 



any point Xo, yo, and using the Laplace equation, we have the theorem: 



the projection of the section of a tangent plane to the surface C = 0i (s, y) 



upon the x, y-plane is a curve having a double point at Xo, yo with real, 



orthogonal tangents, and hence the surface is hyperbolic at every point. 



C = k gives lines of level on C - ^i (x, y), while C^ka in C=i02 (x> 7) 

 gives cylinders which intersect C = ^1 (x, y) in curves of quickest descent. 



The second part of the paper deals with the linear fractional function 



Zj = which has the fundamental invariant points fi,lo about 



which a canonical transformation may be constructed so that Z ^= o, when 



Zi — A « — y/*! /Z — f,\ 

 Z' :=,! I ; Z = ac , Z' =/2. This function is Z ^ j\'_rT ^ ^^T 7( \ /"UTT /• 



z — /"i z — /•, 



The modulus of y ^ , and amplitude of y V, set, respectively, equal to 



constants give an elliptic system and an hyperbolic system of circles about 

 and through the two points ,/\, A- Now the transformation 

 Z^ —.A « ->/, /Z-./V 



z = 



?/. /Z-.A\ 



Z2 — Jz 

 sets up a motion about .A, .A which is determined by the modulus 



and the amplitude of "lHJJj^. If mod. =f 1 and amp = o, motion 



« — J J'o 



r52(f> _ _ ,52$ 



*Where > — r denotes the second p irtial of il) with regard to x, and so for -? — r. 



