256 THIRD REPORT — 1833. 



w TT , . 2 tn: . 3 tin . 1^-K 



If any assigned coefficient a^ be required to be determined 

 from this system of equations, we must multiply * them seve- 

 rally by 



n , m It ^ . 2mTr ^ . Smrr _. nmv 



2 sm — —T, 2 sin — --r, 2 sin ^ , ... 2 sin r, 



n + r n + l n + I n + I 



when all the coefficients except a^ will disappear from the sum 

 of the resulting equations : and we shall thus find 



2 f . OTTT . . 2mTt . nrmr'X 



«», = — r~\\ Vi sin — --r + Vo sin — —^ + . • . -/„ sin — -— ;- >. 

 M + 1 L n + \ ^^ « + 1 "^ n + IJ 



It would thus appear that it is always possible to determine a 

 series of sines of n terms with finite and determinate coeffi- 

 cients, which shall be the equation of a curve which shall have 

 n points in common with the curve whose equation is ?/ =: <p x, 

 within the limits corresponding to values of x between and v ; 

 and it is obvious that the greater the number of those points, 

 the more intimate would be the contact of these two curves 

 throughout the finite space corresponding to those limits. If 

 we should further suppose the number of those points to be- 

 come infinitely great, then the number of terms of the trigono- 

 metrical series would be infinite likewise, and the coincidence of 

 the curve which it expresses with the curve whose equation is 

 y =s <p X, would be complete within those limits only, producing 

 a species of contact to which the texxn. finite osculation has been 

 applied by Fourier f . Beyond those limits the curves would 

 have no necessary relation to each other. 



It would follow, also, from the preceding view of the theory of 

 finite osculations, that the curve expressed hy y = <p x might be 

 perfectly arbitrary, continuous, or discontinuous. Thus, it might 

 express the sides of a triangle, or of a polygon, or of a multi- 

 lateral curve, or of any succession of points connected by any 

 conceivable law ; for in all cases when the corresponding or- 

 dinates of equidistant points are finite, we shall be enabled to 

 determine values of the coefficients a^ which are finite or zero 

 by the process which has been pointed out above. 



* This is the process proposed by Lagrange in his "Theorie du Son," in 

 the third volume of the Turin Memoirs, as stated by Poisson in his memoir 

 on Periodic Series^ &c., in the 19th cahier of the Journal de I'Ecole Polytech- 

 nique. 



f TMorie de la Chaleur, page 250. 



