4 The Astronomer Royal on the 



In forming the sine of this quantity we cannot, as before, 

 proceed by expansion in powers of c, neglecting all after the 

 first; and the reason for this prohibition deserves attention, as 

 showing what extreme care is necessary, in a process of ap- 

 proximation, to adopt no method at any one step without due 

 consideration of the effect of rejected terms at that step. Sup- 

 pose, for instance, the terms depending on higher powers of c 



in the expression for ^ Va* + r/ were tfoo'^ P*'"^ °^ *^^' 



2 TV • Ct C • COS w 



depending on the first power, and suppose that —^—===- 



were nearly equal to 10 tt. The rejection of the terms de- 

 pending on the higher powers would only produce in this an- 

 gle an error of — — , which would be wholly unimportant in 

 these investigations. But the expansion of 



sm 



{2 TT , 2 TT . a c . cos n 

 IT ^"' + "• + ICVWT^P 



In a series proceeding by powers of c, would give a succession 



.,,,.. , 2 7rcccos9 



of rapidly diverging terms when ^ ^ ^ had any such 



value as 10 tt; and the expression produced by retaining the 

 first and rejecting the higher powers would not in any degree 

 represent the whole. From this point, therefore, we must use 

 methods which do not imply the rejection of any powers of c. 

 For convenience, put 



- 2'rr ac , - Stt^c 



<^i f"^- T-T77^=7=a' and e, for 



Then 

 and 



\ s/k' + ay " ^ \nf + 6^ 



2 7r ./ .. . c 2 7r 



tJL Vh^ + r^ =yL^/h^-^ a^ + e, cos 9, 



A A, 



2 7r ,To- -o . 2 7r 



'/h'^ + rf = sin-r— »/ J^ + a^ . cos {e. cos i 

 X A, 



+ cos^ \/h^ + a' ' sin {e,cos^). 



A. 



Similarly, 



sin ^ a/ A^ + r,f = smll. »/ h'' + b~^ • cos {e„ cos 

 A A 



27r 



+ cos—- \//i^ + U^ . sin [e,, cos! 



