REPORT ON CERTAIN BRANCHES OF ANALYSIS. 255 



It 



T 



71" 



axis of X at the distance -j- above it : the line whose equation is 



TT 



— -J-, is also parallel to the axis of x, at the distance -j- below 



it : the line whose equation is 



y = cos X «- cos S X + -^ cos 5 x — &c. 



consists of disco7itinuous portions of the first and second of 

 those lines, whose lengths are severally equal to tt. The values 



of y at the points B and b, corresponding to a- = -^ and — ^, 



are equal to srero, since the equidistant points D and C, c and d, 

 are common to both equations at those points. 



It would appear, therefore, in the cases just examined, that 

 the conversion of one member of the equation of a line into a 

 series of sines and cosines would change the character of that 

 equation from being continuous to discontinuous, the coinci- 

 dence of the two equations only existing throughout the ex- 

 tent of one complete period of circiUation of the trigonometrical 

 series : and more generally, if, in any other case, we could ef- 

 fect this conversion of one member of the equation of a curve 

 into a series of sines or cosines, it is obvious that the second 

 equation must be disco7iti7iuous, and that the coincidence 

 would take place only throughout one period of circulation, 



whether from to tt or from — -^ to -^. It remains therefore 



to consider whether such a conversion is generally practicable. 

 Let us take ti equidistant points in the axis of the curve 

 whose equation is 7/ = 4> x, between the limits and tt, those 

 limits being excluded : if we denominate the corresponding 

 values of the ordinate by i/^, i/c^, . . . . i/„, and if it be proposed 

 to express the values of these ordinates by means of a series 

 of sines (of 7i terms) such as 



o, sin a; + ag sin 2 x + «3 sin 3 4? -f- . . . . + a„ sin 71 x, 



then we shall get the following n equations to determine the n 

 coefficients a^, %> ^3 • • • • ^«* 



TT 



yj = ttj sm + «2 sm 



^2 = «i sin -^ — ^ + O2 sin 



y^ = a^ sin - + flTg sm 



