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II. 



NOTE ON THE USE OF CONJUGATE FUNCTIONS 

 IN SOME DYNAMICAL PEOBLEMS. 



By H. C. PLUMMER, MA. 



Read January 26. Published March 6, 1914. 



1. It has been shown by Bohlin (Bull. Astr. xxviii, p. 113) that, by means of 

 the transformation 



x + iy = at?, x - iy = arf, (i 2 = - 1), 



with a corresponding change of the time variable, the problem of two bodies 

 under mutual attraction according to the gravitational law is reduced to the 

 problem of motion under a force proportional to the distance. 



2. The above transformation is imaginary ; but it suggests the real 

 transformation in terms of conjugate functions 



x + iy = (£ + ir\f 

 or x = £ 2 - rr, y = 2^, 



to which correspond in polar coordinates 



r = p", 6 = 2<j>. 

 The kinetic energy is 



T = i(x* + f) = 2(^ +l f)(? 2 + 0> 

 and the potential energy is 



so that the equation of energy is 



2(^ + ^)(r + f) - ii(V + fflr l -h. 



The Lagrangean equation corresponding to E, is 



4 j t |(p + „*)£} - 4?(f° + if) = - 2f,l (^ + n*y, 

 which becomes, in virtue of the preceding equation, 



If, then, the time variable t is changed to r, where 



dt = (g 8 + tf) dr = rdr, 



fS.I.A. fBOO., VOL. XXXII, SECT. A. [2] 



