316 REPORT 1861. 



and a„+2/3o04-a i0'=O is the quadratic equation determining the quotient /z^, 

 in which we suppose for simplicity that a^ is positive. 



If, in particular, we consider the quadratic equation 0^ — D— 0, or rather 

 a=— D — 2a0+0^=O, where a^<D<:(a + l)', we have »i = l, fj^=2a, and we 

 lind, by the symmetry of the period in this case, 



U^=(a'2» Fa' • • • H2k' 2«, H2> H2k)x-v 



which are Euler's formulae for the solution of the equation T^— DT]^=1. 



(ii.) We have already observed (art. 90) that when T^ and Uj are known, 

 T^ and U^^ are defined by the equation 



711 L '« J * 



Either from this equation, or from the cumulantive formulae for T^, Uj., 

 we infer that T^ and U^, satisfy the equation of finite difl'erences, 



2T 



Vx+2— -Vx+l + Vx = Oi 



m 



so that the two series, of which T^ and U,, are the general terms, are each a 



2T 

 recurring series, the scale of relation being 1, — — S 1. 



It is convenient to observe that T_i.= Tj..; but U_x=— Ua;. 

 (iii.) If we denote by \|/ the imaginary arc 



1 log (TrMWD), 



., , T , U,VD . , Tx , \5^VD .' , 



we have evidently --i=co3 4/, -^—. — =&m\^, — =cosa;»f', ^- =smx\l>. 



•'771 nil 7n mt 



The analogy implied by these formulae enables us to transform many trigono- 

 metrical identities into formula3 containing Tx and U^. For example, from 

 the formulae cos ((/) + 0)=cos^ cos + sin ^ sin 0, sin (o!> + 0)=sin cos0 + 

 sin cos <l>, we have, putting ^=^»//, 6=i/\p, where x and y are any positive 

 or negative integers, 



Tx±^=i[TxT3,±DUxUy], 

 (iv.) It is also found that 



I-.=(-i)=^Vt., -?l,?l, . . . (_,)-. 21.). 



Ttl \m 771 m 771 / 



Ui "^ \77l' JTl ^ ^ Til J 



(v.) If g be any integral number whatever, we can always find a solution 

 [T;^, Up,] satisfying the congruences T;^ = T„=?h, mod q, and U;^=Uo=0, 

 mod q. If [T^, U^,] be the least solution satisfying these congruences, \ will 

 be less than 2q, and the residues (mod g) of the terms of the two series T, 

 and U will each form a period of \ terms, so that we shall always have 

 T,+„/^ T„ U,+,.,= U„ mod y. 



