( «75 ) 



and others tlie form 



2 cos n t cos n t , [2 n ] f -5" ti \ = 2 q^ cos n t cos n t % § .. (19) 



A^ L' X 1 1' '''/,' rC fc IC fC 



As has ah'eady been said, the time O- contains a very hirgc mimhei' 



2.T 



of periods — . A certain vahie ot cos nt, once occui-rin"- in the series 

 n 



cos II t^, cos II t^, cos nt^, . . . may therefore be snpposed to repeat itself 



many times. Also, one and the same value of the product cos n tj. cos n tk' 



may be said to occur for many diiFerent vahies of /• and k'. 



Such a product \vill therefore have to be multipUed by very different 



expressions of the form ^k^k', ^^^^h since the different values of | are 



mutually independent, the number of cases in which |a- and |/^' have 



ojiposite signs will be equal to tliat in which they have the same 



sign. It appears in this way that tiie terms (19) will cancel each 



other in the sum. It is only the terms of the form (18) that remain, 



and wo shall have 



a' =- ^—:S [cos' Ht.^n (20) 



§ 9. Here we may begin by taking together those terms in which 

 cos II tic has one and the same value. Let the nnmber of these be 

 Q. Then, we have to repeat Q times the construction of the diagram 

 of velocities, and it may be asked in ho^v many of these Q cases 

 § will lie between given limits % and i,-\-d%, or, what amounts to 

 the same thing, what is the probability for | falling l)et\veen these limits. 



This question may be reduced to a simpler problem. A series of 

 planes, perpendicidar to () X and at equal distances from one another, 

 ^vill divide the spherical snrface into equal j)arts. Therefore, instead 

 of distributing the points D on the surface in an irregular, arbitrarily 

 chosen manner, we may as well immediately distribute the points 

 D' ,at random oxer the diameter, ^vithout giving any preference 

 to one part of the line over another. The })r()babilily in (pieslion is 

 llius found to be ^) 



Hence, among the Q terms in the sum, occurrijig in (20), for 

 which the factor cos' nti, has equal values, there will be QPd% 

 terms, wliicii may be said to have the same |a. Together, they will 

 coidribute to the sum the amouid 



^0 See §§ 13-15. 



46 



Proceedings Royal Acad. Amsterdam. Vul. V. 



