431 



so that Z 



and f,=elta(e,g) 



(read elliptic quotient radius and elliptic true anomaly), are known 



functions of e, g. Moreover x denotes the periodic part of -, and y 



a 



the equation of the centre or periodic part of/; so that 



and x y y are also known functions of e, g. 



Formulae for the development in multiple cosines or sines up to 

 the terms in e* of 



<*' O 



where j is an indeterminate symbol, are given by Leverrier in the 

 ' Annales de 1'Observatoire de Paris,' t. i. (1855), pp. 346-348 ; and 

 what has been done is the deduction from these of the developments 

 in the like form of various functions of the forms 



where,; has given integer values. It is to be remarked that a cosine 

 series is in general represented in the form 2 [cos] 1 cos ig, where * 

 extends from oo to + oo , and the coefficients [cos] 1 satisfy the con- 

 dition [cos] ~ l = [cos] 1 , and that a sine series is represented in 

 the form S [sin]* sin ^r, where * extends from ooto + oo , and the 

 coefficients [sin] 1 satisfy the condition [sin] ~ l = [sin] 1 ' (this implies 



[sin]=0). In the case of a pair of corresponding functions, x m cosjf 



( r \m / r \m 



-) cosjf/and I - J shy/, one of them expanded 



in the form S [cos] 1 cos ig t and the other in the form 2 [sin] 1 sin ig, 

 the sums and differences of the corresponding coefficients [cos]', 

 [sin] 1 (represented by the notation [cos + sin]*, and which are ob- 

 viously such that [cos + sin] ~ l = [cos sin] *, [cos + sin] = [cos] ) 

 are for many purposes equally useful with the coefficients [cos]*, [sin] 1 '* 

 and they are in the memoir tabulated accordingly ; and the several 

 functions tabulated are as follows : viz. 



VOL. x, 2 IJ 



