INVERSE METHOD OF DEFINITE INTEGRALS. 347 



SECTION VII. 

 On Transient Functions. 



22. Let ^ (h, t) be such that when h has a particular value as- 

 signed, the whole function vanishes whatever may be the value of t, 

 except in one case ; (/^, t) under those circumstances, is a transient 

 function having only a momentary existence. 



Thus the function _ , (^ —0.t\l.hH^ ' ^^*^" ^' ^^ P"^ equal to 

 unity is a transient function, because its value is zero in every case 

 except when t = 0, for then it becomes t- — j-^ when h is put equal 

 to 1, that is, it acquires momentarily an infinite value. 



If the value of the function had been always zero, its definite inte- 

 gral relative to t would also be zero; but if we actually integrate from 

 ^ = to t = \ without previously assigning a particular value to h, the 

 definite integral 



2A \\-h \+h\~ ' 



thus this integral is independent of h, and therefore remains the same 

 when h=\, that is, for the transient function. 



By the principles of the Second Memoir we can always form a 

 self-reciprocal function in which the general term may be of any par- 

 ticular kind ; thus if f{t, n) were the type of the general term, and 

 if we put generally, 



Fit, n) = a,f{t, 0)+a,f{t, l) + a^f{t, 2)+ + a„fit, n), 



lastly, if we determine the coefficients a,, «2, a„ in terms of «„ and n, 



by the n equations (arising from the definite integrals) following, 



f,F{i,n).fit,0)=0, 

 Vol. V. Part III. Zz 



