SECT. III.] REMARKS OX THE SERIES. 145 



SECTION III. 

 Remarks on these series. 



179. We may look at the same equations from another point 

 of view, and prove directly the equation 



7 = cos x - cos 3.r 4- - cos o.x ^ cos 7x + Q cos 9# &c. 

 The case where x is nothing is verified by Leibnitz series, 



7T 1 11 11 



7 =1 - 7, ; + ^ - T= + 7: - &C. 

 4 3 o / 9 



We shall next assume that the number of terms of the series 



cos x ^ cos 3# + -^ cos 5o: ^ cos fa + &c. 

 o o / 



instead of being infinite is finite and equal to m. We shall con 

 sider the value of the finite series to be a function of x and m. 

 We shall express this function by a series arranged according to 

 negative powers of m; and it will be found that the value of 

 the function approaches more nearly to being constant and inde 

 pendent of x, as the number m becomes greater. 



Let y be the function required, which is given by the equation 



y = cosx- Q cos 3. + - cos ox-^ cos 7x+...-- -cos (2wi l)a?, 



o o / Jim 1 



7?i, the number of terms, being supposed even. This equation 

 differentiated with respect to x gives 



-r- = sin x sin 3# + sin ox sin 7x + ... 



+ sin (2??i 3) x sin (2wi 1) x ; 

 multiplying by 2 sin Zx, we have 



2 -y- sin 2# = 2 sin # sin 2# 2 sin 3j? sin 2# + 2 sin 5# sin 2^ ... 

 cfo 



+ 2 sin (2m - 3) or sin 2,z - 2 sin (2w - 1) x sin 2#. 



F. H. 10 



