6 The Astronomer Royal on the 



the integrals being taken from 6 = to 6 = 2 tt. Then the 



^ . ^ . 2irvt . 



coefficient ot sm — r — is 



\ . (e,, . sin -^ \^h^ + b^ - E^ . sin-^ ^ K^ + oTj = S. 



rr,, ^ . P 2 TT U ^ . 



The coefficient ot cos — r- — is 

 X. (e„. cos 1^ Vl^Tb^ - E,. cos ^ v/FT^^)= C. 

 and if the two terms be aggregated into one of the form 



P.„(i^.Q). 



it is easily seen that P or the amplitude of vibration at the 

 point in question is ^^S^ + C^ ; and if the intensity of light 

 be assumed proportional to PS it may be represented by 



S2 + C2 

 -^,or 



E^« + E/,2 _ 2E,. E,,.cos ^.(VWTI?- \/¥T^\ 



Our attention must now be directed to finding the numeri- 

 cal value of the function E for different values of e. 

 By the usual expansion, 



e^cos^fl e'*cos*9 e^cos^fl , . 



cos {e cos9) = l - -j-^- + y:^^^ -1.2.3.4.5.6"^ ^''" 



a series which is always convergent, and may therefore be 

 safely used for numerical computation. 



Integrating separately each term, from 9 = to 9 = 2 tt, 

 and observing that if cos- "6 be expressed by simple powers 

 of cosines of multiple arcs, the constant term is 



J_ 1-3-5-7...2?? — 1 

 2" 1.2.3.4 ...» ' • 

 we obtain 



y;cos(ecos9) = 2.x|l-^-|^,+ ^-2^.-^27^ 



and therefore 



e* e* e 



^ ~ ^ "" {2)^'^ (2:4?^ (2.4. 6)» "^ ^^- 



