14 Proceedings of the Royal Irish Academy. 



The conditions which most naturally present themselves in physical 

 problems are that, when each element of the sum, or integral, is multiplied 

 by the corresponding value of e""^', the initial values of the sum and of 

 its differential coefficient with respect to time shall each lie functions of x 

 which are arbitrarily assigned.* 



Art. 3. The Expansion in Trigonometrical Series modified so as to refer to 

 Two Arbitrary Functions. 



Seeking first, as being of most interest, the Trigonometrical Sum theorem 

 which ie now required, we have the following problem, modified from that of 

 Part II., Ai-t. 2 :— 



It is required to have simultaneously,! when t = 0, 



S (^e"^ + ^e-f'^) e^'i = ^ (a;), (1) 



d/dt . 2 (^e"' + ^e-M^) e""^' = ^ (x), (2) 



\p, X being arbitrary functions, where the characteristic values of £/A and 

 of n are given by the equations 



Aei^"F, („, a) + Be-i^^F^ (- fi, a) = 0, (3) 



Ae>^''F, {fx, h) + Be-'^oF, (- fi, h) = 0, (4) 



the F's, denoting polynomials in fi of which Fi, F, are of the same order p and 

 Fi, Fi of the same order q. 



Unless the F's have reference to some physical problem, the solution 

 in so far as the satisfaction of (2) is concerned appears to be in general 

 impossible. But an expansion which satisfies (I), (2), and which, whenever 

 it contains no term arising from a residue at /i = 0,J is of the desired type, 

 is given by the equation 



{■inir' 



c. 



d^ r i^;-»)ii;(- fx) - e->^i^^)F,(,i) j 



I' {6"'i"-'')i?',(-;u) - e-i^C-'-)F,(u)\ \^P{u) + {cm-\{u)}du 



+ Terms of index {p - 1) and (^j - 2) in j Z', (- fi) \/jr^\p{a) - fi-^'{a) + c"V"'x(«)l 



+ F, (ju) i/x-'^K^) + /^-^'(«) + c-v--x(«)l } [ 



- \&^i^-'')F.,{- u) - e-^l^-iiXi")} (Terms of index {q - 1) and {q - 2) in 



{F,{- ^) |^->^(J) - !_c^'{h) + e-^ir-x{h)\ -r F,(j,) l^->^(5) + ^-f (&) + 6-V=x(*)}j 



• The conditions might, however, assume other forms ; for example, one might refer to some 

 Oilier time. 



t We may, if we prefer it, replace (1), (2) by the simultaneous equations 2Aei^' = d{x 

 2B«-»" = (j)(i-), 9. «) being arbitrary. 



I This limitation does not affect the problem as stated in the preceding fool-note. 



