SECT. VI.] REMARKS ON THE SERIES. 207 



1st. The series arranged according to sines or cosines of mul 

 tiple arcs are always convergent ; that is to say, on giving to the 

 variable any value whatever that is not imaginary, the sum of the 

 terms converges more and more to a single fixed limit, which is 

 the value of the developed function. 



2nd. If we have the expression of a function f(x) which cor 

 responds to a given series 



a + b cos x + c cos 2x + d cos 3# + e cos 4# + &c., 

 and that of another function &amp;lt;/&amp;gt; (a?), whose given development is 



Q.+ ft cos x + 7 cos Zx + 8 cos 3x + e cos 4?x -f &c., 

 it is easy to find in real terms the sum of the compound series 



act + b/3 + cy -f dS + ee + &C., 1 

 and more generally that of the series 



ax + 6/3 cos x + cy cos 2# + cZS cos 3# + ee cos 4tx + &c., 

 which is formed by comparing term by term the two given series. 

 This remark applies to any number of series. 



3rd. The series (5^) (Art. 234s) which gives the development 

 of a function F (x) in a series of sines and cosines of multiple arcs, 

 may be arranged under the form 



+ cos x \ F(a) cos ado. + cos 2# I F (a) cos 2s&amp;gt;cZa -f &c. 

 + sin x I F (a) sin acZa + sin 2x I F (a) sin 2a dx + &c. 



a being a new variable which disappears after the integrations. 

 We have then 



+ cos x cos a + cos 2x cos 2a + cos 3# cos 3a + &c. 

 + sin cc sin a + sin 2x sin 2a + sin Sx sin 3a + &c. , 



1 We shall have 



fir 



f 



Jo 



t(x)&amp;lt;f&amp;gt;(x)dx=CMT+lT{bp+Cy+...}. [R. L. E.] 



