93 



CoNJU(iATE Functions and Canonical Transformations. 



By David A. Rothrock. 



(Abstract.) 



It is known that any function, (p [Z), of a complex variable, Z = x -\- 

 i J, may be separated into a real part <pi (x, y an imaginary part, I o., x, y), 

 and that ©1,0, ^ach satisfy Laplace's equation i^ -f- - — '- = o * A very 



f)X, ('J 2 



elegant geometric interpretation of these two functions Oj, o, may be had 

 by equating each to a third variable ^ : ^1 (x, y) = C, 62 (x, y) := C. Each 

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

 is true. By developing : — Oi (k, j) 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 = Oj (x, 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. 



: = k gives lines of level on C — <Ai (x, y), while C^kg in C = 02 (^» J) 

 gives cylinders which intersect C = <?i (x, y) in curves of quickest descent. 



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



Zi = — which has tlie fundamental invariant points f^, /', about 



y ; + <^ 



which a canonical transformation may be constructed so thafe Z := o, when 



Z' — /", « — / /"i /Z — /", \ 

 Z' =J\ ; Z = -X , Z =/,. This function is Z = ^, _y = ^^— „ ^^ l Z^^^ / ' 



y r 7 r 



Themodulusof^":^-^^. and amplitude of 2^^"V. set, respectively, equal to 

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

 and through the two points ./'i. ./V Now the transformation 



Z, -/, - a-y/,\Z -fj' 

 sets up a motion about ./'i,/, which is determined by the modulus 



and the amplitude of ^'^ ~ '' ■' ' . If mod. =f 1 and amp = o, motion 



« — / /2 



''Where , — r denotes the second n irtial of <I> with regard to x, and so for ^ — r. 

 (5x- "y^ 



