122 MEMOIRS OF TnE NATIONAL ACADEMY OE SCIENCES. 



while P is tlic ])erio(l of vibration. For any otlier element of OA a velocity is imparted \vlii(-li 

 bears the same ratio to 



...sin2.{i_^|, 

 ■where S is the difference of the distances of tlie two elements considered from p. If the distance 



,v,2 



Oj) is represented by E. 6 is evidently e<|unl to ' . Tlie total velocity given to the particle at p 



will equal the sum of the velocities due to each element, whence we may write for tlic iiortion 

 included between x' and .*", 



This developed gives 



V CO sin 2Tt ' \ cos 2 ;r - ' , f7,r— cos ^n I sin 2 tt -^ dx 

 VJx' li U A. PJx' 2EA. 



The intensity of illumination at jj is proportional to the quantity of energy transmitted to it in a 

 given time, and this is obviously proportional to the mean square of r, but since the mean value 



of sin 2 7r cos 2 ;r is zejo, while that of sin^ 2 ;r ^ is a constant e(iual to tliat of cos'^ 2 tt „ we 

 find for an exjiression for the intensity 



In calculating the quantity of light due to any point on the sun, we must extend tlie integration 



from a value of .r' corresponding to the limit of the moon up to inlinity. But if the value of ,t' be 



III A 

 but a few times greater thau-^ ^ the value of I becomes indefinitely small. The relation of j,' to 



the apparent angular distance from the moon's limb is, if we represent this angle by n, given by 

 the equation 



X 



whence we conclude that in order that any portion of the sun's surt^ice may give a sensible 

 amount of light to p, its angular distance from the moon's limb shall be not many times greater 



than -\/^„; in other words, the distance within the geometrical shadow of the moon, where light 



due to diffraction may be found, is limited to inches rather than miles. Thus it would appear that 

 the corona can have no explanation founded upon this iihenomenon of light; and, indeed, there 

 would be no escaping the conclusion if the suppositions at the base of the argument are correct. 



But, in setting the phase of vibration equal -^^^ , we have tacitly assumed that the motions in the 



2KA 



different wave-surfaces are strictly similar; this is not the case, for we know that two surfaces difter- 



ing by a few tens of thousands of wave-periods iu their origin are wholly incapable of producing 



phenomena of interference. For example, we can secure Newton's rings with pioper precautions as 



to the homogeneity of the light with a dift'erence of jjath of half an inch, but no phenomenon of the 



kind can possibly be discovered if the difference is as great as 2 inches. Accordingly we can only 



find the value of I by taking the mean value of 



[X '•■•-(r-^'-'H 



