Lord Rayleigh's Investigations in Optics. 409 



The intensity may be represented by 



[2 f 'cos (hr*)rdr¥+ [2 Tsin (h^)rdr] 2 , 

 Jo Jo 



the scale being such that the intensity is unity in the case of 

 no aberration (A = 0). As before, we find 



2j o ^^ = ^|l^ T+ ^-^ I ^ + ...j, 

 whence 



+ sin A {¥ - e^Sri + • • •}- • < 9 > 



- cosA t^~67io7u + ---)- • (10) 



Thus, when h — \ir, 



.f> /! 4x 7 1*32945 ^.f 1 . „ A , -35424 



2 1 cos ( i-rrr^) r dr = — — > 2 1 sin (i irr) r dr == — -~- ; 



Jo v 2 Jo v £ 



[2r i cos(i7rr 4 )r^] 2 4-[2r , sin(l7rr 4 )^^] 2 = -9464. 



Jo Jo 



Again, when h = ^ir, 



2 Tecs (i7rr 4 )rdr = '77989, 2 Tsin (I'm- 4 ) r<fr = -43828, 



Jo Jo 



[2 f 'cos (l7rr 4 )r ^r] 2 + [2 f 'sin (ivrr 4 ) y ^] 2 ='8003. 



Jo *. 



Again, when h = 7r, 



2 i 'cos (irr A ) r dr = -3740, 2 (sin (tt/- 4 ) 7' dr = -5048, 

 *J o Jo 



[2 f'cos(7rr 4 ) r drj+ [2 f 'sin (tit 4 ) rdr] 2 = -3947. 



Jo Jo 



Hence in this case, as in the preceding, we may consider that 

 aberration begins to be decidedly prejudicial when the wave- 

 surface deviates from its proper place by about a quarter of a 

 wave-length. 



As an application of this result, let us investigate what 

 amount of temperature-disturbance in the tube of a telescope 

 may be expected to impair definition. According to the ex- 

 periments of Biot and Arago, the refractive index //, for air at 



