Periodicities when the Periods are unknown. 371 

 1st remainder =u x sin (jJ 1 + m ^r b^\^± — ^2- 



2nd remainder=zi 1 sin ( IL+ — — -r^ + fr^ ^Sl — LI 

 \ 2 / 2i r 



+ % sin^U 2 + ? ^r& 2 + ^^^ , + J &c. 

 3rd remainder =u x sin fjJ 1 -\- — i _ r& 1 + 2& 1 ^ lik—ll 



+ ti, sin f TJ 2 +^j r&o + 25 2 ) " 1)r + , &c. 



+ % sin {u 2 + ^-^r-l^ 2 }li^l r + , &c. 



Pursuing the mode of entry adopted for the first means and first 

 remainders, the rth means will occupy the 2rth line of our table and 

 the rth remainders the (2r + l)th line. 



In future we shall for shortness write — 



U l + ^ r& l = V l ^ U 2 + ^ **8= V* &C. 

 2 2 



14. The following calculations are made for ready reference, and 

 to show how the factors of the form — — and — — — which 

 occurred in our rth means and rth remainders are obtained — 



The factor for — 



1st means =1 



1 



1st remainders = 1—1=^ — -. 



q q 



2nd means =- — -. 



r 



2nd remainders^illi-g^l=^- 1 )g-^- 1) = r g- 1) ^- 1) 



q f 2 » q* 



