SECT. IV.] SECONDARY INTEGRAL OF LINEAR EQUATION. 423 



It is required to discover this second integral of the proposed 

 equation, which cannot be more general than the preceding, 

 but which contains two arbitrary functions. 



We can arrive at it by summing each of the two series which 

 enter into equation (X). Now it is evident that if we knew, in 

 the form of a function of x and t, the sum of the first series which 

 contains f(t), it would be necessary, after having multiplied it by 

 dx, to take the integral with respect to x, and to change f (t) into 

 F (t). We should thus find the second series. Further, it would 

 be enough to ascertain the sum of the odd terms which enter into 

 the first series : for, denoting this sum by /i, and the sum of all 

 the other terms by v, we have evidently 



[* [* dp 

 = I ax \ dx -j- . 

 Jo Jo 



It remains then to find the value of p. Now the function 

 f(t) may be thus expressed, by means of the general equation (B\ 



It is easy to deduce from this the values of the functions 



It is evident that differentiation is equivalent to writing in 

 the second member of equation (5), under the sign I dp, the 

 respective factors p 2 , +p*, p 6 , &c. 



We have then, on writing once the common factor cos (ptpz), 



Thus the problem consists in finding the sum of the series 

 which enters into the second member, which presents no difficulty. 

 In fact, if y be the value of this series, we conclude 



&amp;lt;?y 2 , p 4 ^ p 6 ^ 8 , i &amp;lt;2*y 5 



^=-/+- or s? = ~ py - 



