374 Mr. V. N. Nene. On a Method of Tracing 



(p + l)th mean from the second set of means 



= u Y sin (V 1 + i h sin ( Y 2 +p& 2 ) 2^ + , & c . 

 (p-f-l)th mean from the third set of means 



=Uj sin (Yi+Jp&O (gl ~ 1)2 + ^sin (Yo+^,) (g ~ ~ 1)3 + , &c. 



Qj + l)th mean from the rth set of means 



= Uj sin (V 1 + pbj (jizjj^ + % sin (Y 2 +i?& 2 ) 1 + , &c. 



2l' 2/ 



(p + l)th remainder from the rth set of remainders 



=«! sin (Yj fpk) ^ sin (Y +»fc 2 ) i^—^— + , &c. 



And therefore if we nse a variable integral number P (which may 

 be zero) as a multiple to the angle of the type b, we can make one 

 general expression serve to represent the numbers of any one horizontal 

 line; this will be for the 1st, 3rd, 5th, &c, lines as follows : — 



1st line, observation series 



= u +■ {u Y sin (Y 1 + P&J } + {u, sin (Y 2 + P& 3 ) } + , &c. 



3rd line, 1st mean series 



= u+{ u Y sin; ( Y, + P&,) } — + { ih sin (Y 2 + Pi,) }— + , &c. 



2i % 



5th line, 2nd mean series 



= { i h sin ( V 1 + P&x) } + { sin (Y 2 + P6 2 ) } ^q 1 + , &c. 



2i 22^ 



7th line, 3rd mean series 



= K sin ( Y x + P&O } <£il^2! + { a 2 sin (Y 2 + P& 3 ) }{££T _D! + , & c . 



2i W 



(2r + l)th line, rth mean series 



= {u, sin (V 1 + P& x ) J^-^ll^. 1 + sin (Yo + P6.0 i fe" 1 ^'" 1 + , &e. 

 (2r + 2)th line, rth series of remainders 



K sin ( Y x + P&O } (gl ~ 1)r + { ti, sin ( Yo + P&o) } (g2 ~ 1)r + , &c. 

 Where when P is put =0, 1, 2, &c, successively the expression in 



