96 The Rev. B. Bronwin on a particular Transformation 

 we find 



? dP mh8 dr dv p 3 dt* + p 2 V/3 2 ^ dt* ) + dr 



ih(3 dr dv p 3 dP ' p 2 \/3 



Make /3= l+p, r=t + z. In substituting these values in 

 (1.) and (2.), we will, for simplicity, neglect the second and 

 higher powers of the disturbing force. And thus, observing 

 that 6, p, and z are of the order of that force, we shall find 



dz „ 1 rdR 



dt 

 and 



1 /*dR 1± 



9 dt 2 p 2 \ P * dt) p 3 m*h dr du 



dR 



+ -r = 0. 



Or 



dr 



d?p fu_ GA* IdR l_didK 



dP + P 3P f + p dr m 2 hp dr du 

 gfi fdR 



rrfihp 3 J du 



> • (*•) 



which will give z and p in terms of t. 



In estimating the effect of the higher powers of the disturb- 

 ing force, we must take account of the powers of p and z which 

 have been neglected. And in expanding R, we must make 

 r=p(l+p), v = mu, and substitute for p and u their known 

 values in functions of t. But for the first power of the dis- 

 turbing force, we must put t for t, and makep = 0. And for 



-.i - ^ dR . dR 



this first power, we may change -r- into -7-, operating on e, 



the epocha, only where it stands alone without ■or. And we 



i , dR . a dR . 1 dR & , 



may also change — =— into r-, or into - -7—, afterwards 



& dr r da p dp 



making p = 0. Then for the higher powers of the disturbing 



force, we shall only have to change t into t + z by Taylor's 



theorem, and to put for p and z their approximate values as 



they arise. 



We might have made v—u + ctnt instead of v = mu, and have 

 transformed (a.) in exactly the same manner ; and this is Han- 

 sen's method. In the development, we shall easily perceive 

 how to determine the constants £ and m, or a. These would 

 give the progression of the apse. 



By the common method r and v are found in terms of if 

 from the equations 



ddv 2h^ 1 PdR 



dt 



2A R 1 PdR : . 



