118 Sir W. Thomson on the Front and Rear of 



Using now the following notation, 



say 0=(°dd sin 2 ; cay 0=Cd0 cos 2 . (29), 



for two integrals which have been tabulated by Airy* through 

 the range from to 5*5 ,t / -~ we reduce (28) to 



+ [^(V£~ <B v / P +s ^( <B \/|)] sin (j*-" 1 -!) 



- [^(V^ +<B \/?)~ ca K <0 v/i)] C0S (7^ +(U< ~i) 



- [fv(«\/£+'»\/D"" v (*v/f)] sin (^ ,+0>< -£) • • • ( 3 °: 



The interpretation of this is eased by putting it into the form 



<J> = ^ y\ Qcosf — x— cot—e)— Rcos (— x + wt — A I ...(31), 

 where 



q= { jW(*\/£ -»\/f) +ca y(»\/ffi 



E= \/|{ [ ca ^ (* V& +w \/i) - cay (• v/p]* 

 + K (* \/£ + "Vj) - say (« \/$\ * V- (34) ; 



and 



. . _ 1 ^0\/& + "\/i)-^(' B \/f) r .,,, 

 / =tan — 7 7~q TT\ 7 7*\ ~ I • • ' ( 35 >- 



ra nVs +ffl Vj)" fflJ, rvy) 



* ''Tracts " (Undulatory Theory of Optics, last page). 



