SECT. VI.] REMARKS ON THE DEVELOPMENTS. 199 



tinguish clearly the cases in which they represent the general 

 integral from those in which they include only a part. It is 

 necessary above all to assign the values of the constants, and 

 the difficulty of the application consists in the discovery of the 

 coefficients. J^is remarkable that we can express by convergent 

 series, and, as we shalPsee Tn the sequel, by definite integrals, 

 the ordinates of lines and surfaces which arenot subject to a 

 _ continuous law 1 . We see by this that we must admit into analysis 

 functionswKich have equal values, whenever the variable receives 

 any values whatever included between two given limits, even 

 though on substituting in these two functions, instead of the 

 variable, a number included in another interval, the results of 

 the two substitutions are not the same. The functions which 

 enjoy this property are represented by different lines, which 

 coincide in a definite portion only of their course, and offer a 

 singular species of finite osculation. These considerations arise 

 in the calculus of partial differential equations; they throw a new 

 light on this calculus, and serve to facilitate its employment in 

 physical theories. 



231. The two general equations which express the develop 

 ment of any function whatever, in cosines or sines of multiple 

 arcs, give rise to several remarks which explain the true meaning 

 of these theorems, and direct the application of them. 



If in the series 



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



we make the value of x negative, the series remains the same ; it t ^ 

 also preserves its value if we augment the variable by any multiple 

 whatever of the circumference 2?r. Thus in the equation 



- TT&amp;lt; (x) = x I &amp;lt;/&amp;gt; (x) dx -f cos x l(f&amp;gt; (x) cos xdx 



+ cos 2# Iff) (x) cos 2xdx + cos 3# /&amp;lt;/&amp;gt; (x) cos Sxdx + &c....(i/), 



the function $ is periodic, and is represented by a curve composed 

 of a multitude of equal arcs, each of which corresponds to an 



1 Demonstrations have been supplied by Poisson, Deflers, Dirichlet, Dirksen, 

 Bessel, Hamilton, Boole, De Morgan, Stokes. See note, pp. 208, 209. [A. F.] 



