452 Mr. Tovey's Researches in the Undulatory Theoi'y: 



and change, accordingly, the equations (9.) into 

 w^ + s + p S/ = , 



p 



By eliminating p from (12.) we have 



{rC~ + s) K + s') = sf . (13.) 



The sums s , s', S; , comprise only x and quantities depend- 

 ing on the nature of the medium; therefore, the last equation 

 determines the value of x, supposing that of n to be given. 



Whatever be the nature of x, we may, by the principles of 



analysis, suppose x = /z (cos / + -v/ — 1 sin /) : (^^-O 



where li and i are real quantities. Therefore, since 



X A a; x^ A ^^ . x^ A .^•^ „ / 1 - \ 



e.A. = 1 +_, + __ + j-^;^ + &c. (lo.) 



and 



(cos ? + V — 1 . sin ?)w* = cos wiz = V" — 1 .sin ?« z , (16.) 

 (where m is any number,) we have 



h cos « , Iv cos 2 « . „ o 



eKA«_ 1 = — - — A.r H ; — A.f^+&C. 



1 1.2 



(h sin i . /i^ sin 2 / ^ , o \ 



(17.) 



By substituting this development of (e*^ ^ ■* — l ) in the equa- 

 tions (8.), and, for the sake of abridgement, putting 



A = SpAcr, A'=2j3'Aj;, 



•^i — 2 ' ^^ / — 2 * 



_2joA.t-^ ., _ X]/ Aa ^ 



-2- 2.3 ' ^--g-T-g" » 



&c. &c. (18.) 



B = S gf A .r , 



_2yA^ 



B„ = 



_ 2gA 



&c. 

 and a- = Ah cos / + A^ //^ cos 2 / + Ag /^^ cos 3 / + &c. 

 0-' = A' // cos i + A', li" cos 2 « + A'g /^'^ cos 3 ? + &c. 

 <T^— All sin 2 + A/ IC- sin 2 / + Ag A^ sin 3 ? + &c. . 



cr' =A'/isin/ + A',A'sin2/ + A'27i^sin3/ + &c. *^ 



