SECT, IV.] NUMBER OF ARBITRARY FUNCTIONS. 401 



or t; = a + -^ a t + r-r a 4- a + &c. 



|2 - |4 6 



l a .................. (Z). 



O O 



In this series, a and b denote two arbitrary functions of t. 



If in, the series given by equation (^) we put, instead of 

 a and b, two functions &amp;lt;/&amp;gt; (t) and -^ (f), and develope them according 

 to ascending powers of t, we find only a single arbitrary function 

 of x, instead of two functions a and b. We owe this remark to 

 M. Poisson, who has given it in Volume vi. of the Memoires de 

 TEcole Polytechnique, page 110. 



Reciprocally, if in the series expressed by equation (T) we de 

 velope the function c according to powers of x, arranging the 

 result with respect to the same powers of x, the coefficients of 

 these powers are formed of two entirely arbitrary functions of t ; 

 which can be easily verified on making the investigation. 



400. The value of v, developed according to powers of t, 

 ought in fact to contain only one arbitrary function of x ; for the 

 differential equation (a) shews clearly that, if we knew, as a 

 function of #, the value of v which corresponds to t = 0, the 

 other values of the function v which correspond to subsequent 

 values of t, would be determined by this value. 



It is no less evident that the function v, when developed 

 according to ascending powers of x, ought to contain jwo com 

 pletely arbitrary functions of the variable t. In fact the dISerential 



equation -7-3 = -7- shews that, if we knew as a function of t the 



value of v which corresponds to a definite value of x, we could 

 not conclude from it the values of v which correspond to all the 

 other values of x. It would be necessary in addition, to give as 

 a function of t the value of v which corresponds to a second value 

 of x } for example, to that which is infinitely near to the first. All 

 the other states of the function v, that is to say those which corre 

 spond to all the other values of x, would then be determined. The 

 differential equation (a) belongs to a curved surface, the vertical 

 ordinate of any point being v, and the two horizontal co-ordinates 

 F. H. 26 



