524 Prof. Arthur Schuster on 



This represents an oscillation having a maximum velocity 



z 8 sin kI cos pT. 



q 2 -/c 



If the energy contained within a region dX of the original 

 light is B<i\, where B is a constant, the energy contained 

 within the region defined by p and p +dp will be 2'iT^adp/p' 2 . 



The square of the maximum velocity may be taken as a 

 measure of the energy transmitted in a given time. Changing 

 the variable to k by the relation p^a/c/y, and introducing 

 r=ar/y, the total energy E of the separate trains of waves 

 passing through a small area s at the focus of the telescope 

 may, when their number becomes large, be expressed as an 

 integral. To get correct results we must restore the omitted 



factor -x — ^- in the amplitude of the disturbance, and 



writing A=BA 2 cos 2 £/F 2 r , 



8 



7r^"J (g 2 — /e 2 )' 



In order to find the value of this integral, we transform it 

 as follows : — 



P °° q c2 d/c 



E = — As 1 -— — 5T5 sin 2 kI cos 2 icr. 



Jo (? 2 -* 2x2 



sin 2 fcl cos 2 /cr 



'. (?*-« 2 ) 2 



r°° die r°° K^dfc 



= l 7-5 9* sin 2 ^ cos2 *** + \ ~n> 5\9 sin 2 kI cos 2 /c^ 



Jo (? -* ) Jo fa 2 -* 8 ) 2 



f 00 d/e C™ die d 



= J o (^r^y sin2 ** cos2 «■ - * J o (^i^) ^ (* ^ " l cos 2 «■) 



i d fc ■ icdic 



= i l rr - — 9V sin 2 k:Z cos 2 ^ — A I t-3 5- (Z sin 2*;Z cos 2 kt 



Jo (?*-*) J (? 2 -* 2 ) 



— r sin 2/crsin 2 «/). 



The values of these definite integrals are easily found by 

 expanding in Fourier series functions which between and I 

 shall be expressed by sin 2qx or cos 2qx, and vanish for x> I. 

 The general equations which are applicable here are : 



4 C™ C l 

 0(#) = — 1 I <£(V) cos 2kx cos 2/cldtc dX 



-7r Jo Jo 



(/> (X) sin 2*^; sin 2/eZ cfo dX. 



Putting <j>(a) = sm2qa: or cos2g^, with the condition that 

 cosql=l, we obtain the following four equations, which will 

 hold for all values between x=0 and x = l, but not necessarily 

 at the limits. 



