THE GENERAL PERTURBATIONS OF THE MINOR PLANETS. 35 
CHAPTER II. 
Derivation of the Expressions for Bussew’s Functions for the Transformation of 
Trigonometric Series. 
Ge 20 ° . ° 
-The value of (5) given thus far is found expressed in a series of terms the argu- 
ments of which have the eccentric anomaly of the disturbing body as one constituent. 
But as the mean anomaly of both bodies is to be employed, it will be necessary to make 
one transformation ; and the next step will be to develop the necessary formule for this 
purpose. HAwnseEv, in his work entitled Entwickelung des Products einer Potenz des 
Radius Vectors ct cet., has treated the subject of transforming from one anomaly into 
another very fully ; what is here given is based mainly on this work. 
Calling ¢ the Naperian base, and putting 
= GUE. | = Gua". 
we have 
yy’ = (cose + /—1 sin e) (cos e+ /—1 sine’); 
also 
yy” = (eoste+ f—I1 sine’) (cosv’ e+ Y—1 sin?’ é’) 
= cos Ge— 7 e') | f= sin Ge —7 &’'). 
Denoting the cosine and sine coefficients of the angles (¢e—7 «’) by (G75) 
and (7, 2’, s) respectively, the series 
F=>(41,c) cos (¢e—7 & )—=ES V—1 (4,7, 8) sin (¢e —7e’) (1) 
ean be put in the form 
FH=1353 §(47,c) -V—1 2,8) } yy”. (2) 
