REPORT ON CERTAIN BRANCHES OF ANALYSIS. 255 



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

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

 it : the line whose equation is 



y = cos X 5- cos 3 j: + -p- cos 5 x — &c. 



consists of discontinuous 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 zero, 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 circulation 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 discontinuous, and that the coincidence 

 would take place only throughout one period of circulation, 



whether from to tt or from — -p- to -g-. It remains therefore 



to consider whether such a conversion is generally practicable. 

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

 whose equation is y = <^ x, between the limits and tt, those 

 limits being excluded : if we denominate the corresponding 

 values of the ordinate by y^, yc^, . . . . y„, and if it be proposed 

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

 of sines (of n terms) such as 



«i sin ar + Og sin 2 a: + ffg sin 3 /c + . . . . + «„ sin n x, 



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

 coefficients «!, a^, a^ . . . . «„• 



Sr, = «ism^-^+a,smjj-^+«3sm^^ + .«„sm^^-^j, 



Stt, .Gtt .Qtt .3«7r 



2^3 = a, sm ^^-p-j- + a, sm ^-^ + a, sm— ^ + . a„ sm— ^j. 



