434 THEORY OF HEAT. [CHAP, IX. 



variable a. The variable x is only affected by the symbol cosine. 

 It follows from this, that in order to differentiate the function / (x) 

 with respect to x, as many times as we wish, it is sufficient to 

 differentiate the second member with respect to a under the 

 symbol cosine. We then have, denoting by i any integer number 

 whatever, 



We take the upper sign when i is even, and the lower sign 

 when i is odd. Following the same rule relative to the choice 



of sign 



We can also integrate the second member of equation (Z?) 

 several times in succession, with respect to x\ it is sufficient to 

 write in front of the symbol sine or cosine a negative power 

 of p. 



The same remark applies to finite differences and to summa 

 tions denoted by the sign 2, and in general to analytical operations 

 which may be effected upon trigonometrical quantities. The chief 

 characteristic of the theorem in question, is to transfer the general 

 sign of the function to an auxiliary variable, and to place the 

 variable x under the trigonometrical sign. The function f(x) 

 acquires in a manner, by this transformation, all the properties of 

 trigonometrical quantities ; differentiations, integrations, and sum 

 mations of series thus apply to functions in general in the same 

 manner as to exponential trigonometrical functions. For which 

 reason the use of this proposition gives directly the integrals 

 of partial differential equations with constant coefficients. In 

 fact, it is evident that we could satisfy these equations by par 

 ticular exponential values ; and since the theorems of which we 

 are speaking give to the general and arbitrary functions the 

 character of exponential quantities, they lead easily to the expres 

 sion of the complete integrals. 



The same transformation gives also, as we have seen in 

 Art. 413, an easy means of summing infinite series, when these 

 series contain successive differentials, or successive integrals of the 



