Sir Wjlliam Rowan Hamilton on Fluctuating Functions. 309 



For example, we might assume 



p„ = - r do sin 0-^ cos (2a sin B) ; (k''^") 



which would give, by (a"), and (b"), 



N„ = ^ r do sin e sin (2a sin 6) ; {V"') 



M^zz-Tc/i' vers (2a sine); (m^^^^) 



and finally, by (r'), 



z;r = 2r(;0sin0 = 4. {n''"') 



This expression (k^^^^) for p„ satisfies all the conditions of the ninth article; for 



4 

 it is clear that it gives a value to n„ which is always numerically less than - ; and 



7r 

 the equation 



which is of the form (g), is satisfied by all the infinitely many real and unequal 

 roots of the equation 



C f^0cos(2asin(?) = O, (p^^^^) 



which extend from a= — cotoa=GO, and of which the interval between any 

 one and the next following is never greater than w, nor even so great ; because 

 (as it is not difficult to prove) these several roots are contained in alternate or even 

 octants, in such a manner that we may write 



mr TT nit 

 "">-2-4'<T- (1 > 



We may, therefore substitute the expression (k''^") for p, in the formulae (a), 

 (b), (c), &c. ; and we find, by (b), if jp > a, < 6, 



/, = TT-' \ da^ d^ r de sin 0^ cos {2^ (a - x) sin 0}/, ; {v''"') 



^a •^o •^o 



that is, 



