OBii — Extensions of Fourier'' s and the Besset- Fourier Theorems. 17 



And at points between a, h the string is subject to the equation 



cPtpldx" ■■= c-'-d:'<p/dt- ; (12) 



while each displacement involves the time factor c^""'. 



Let the minors of the above-mentioned determinant, Aa, be denoted by 



Let the elements of another column, to be used in forming determinants, 

 and derived* from Aa, be 



+ {M,,fiC- + k,;c) y, + MnCVi + . . ., (13) 

 (M^ific' + /lojc) iji + MniCVi + {M.i.ii.ic" + knc) iji + M,.xVi 



+ {Miifxc- + hiC) 2/3 + M;^cvj + . . ., (14) 

 (Mi.fic- + A-jic) y, + M^iCVt + {M^^nc- + k^^c) y-, + Mi^^cVi 



+ {AlafiC^ 4 k^-iC) 7/3 + M■i^CV^ + . . ., (15) 



in which the y's, denote initial displacements and the v'& initial velocities. 

 And let the determinants obtained by replacing the first, second, etc., columns 

 of Aa by this column-be denoted by nlli, aflj, etc., respectively. 



Let a similar notation apply to the end I, the suffix h being introduced 

 when necessary. 



Also let the initial displacement and velocity at all points of the string be 

 Tp, X, respectively. I suppose that i//, ip', \jj", x, X ^^^ Diriehlet functions, and 

 that ^, i//', x are continuous. 



As a Y'l is identical with fa, it suffices to give the vahies of ^, a ^i, and 4 Fj, 

 the analogue of ^3^2 at the end b. These are 



<j> = - (47^^)-' 



61^'%, 





- {e'^'-''-'-)FtUw, ^i)Fa{^ic, - n) - c^(''-^)Fi{,jic, - fx)Fa{fic, f,) 



(16) 



(47ri)-' 



e^'^^du 



\2fjiT.A, 



{e^C-'Ji^if/iC, -fi)- e-'^("-*)i^j(/iC, ;u)l \^p{u] + {cfi^xhijldii - 2in, 

 + 2all. \ci-^^-'')Fi(uc.,-ij) - e-^i'^-^)Fi{i.iC, /k)U.„| ~ Denominator of (16) 



(17) 



* See Kouth, " Dynamics of a System of Rigid Boilies," Part ii., Art. 366. 

 R.I.A. PROC, VOL. XXIX., SECT. A. [3] 



