Bessel- Clifford Function, and its applications. 523 

 More generally, with a cbange from x=e e to z = e <t> , x — z h y 



6 = kcb, and writing D for tj, 



r d<p 



(D + nk)(D + n - 1 . *) D (D - *)w = e^w ; . . (4) 



and with a change of the dependent variable from w to y, 

 w=z h y = e h< l ) y, 



(A) {B + h + nk)(D + h + n-l.k)(I) + K){B + h-k)y = e 2k '!>y, 



with the solution 



y = e-^tv = z- h C n (z k ) = aT*C»(aO. . . (5) 



Conversely, starting from the D.E. of the whirling or 

 lateral vibration of a bar or tongue, of variable rectangular 

 or elliptic section, varying as z p broad and z q deep, 



< B > a?(* + *S) = "> 



or, with p + 3q = r + 2, p + q = s — 2, z = e < t > , 



Z2 i( zr ' z2 S) = ** D(D-l)«*JD(D-l)y = e**y, (6) 



(C) (D + r)(D + r-l)D(D-l>y = «<*- r ^y ; 



and an identification is to be made between (A), (B), and (0), 

 if (C) is to be solved by the Cliff ord-Bessel function. 



With s = r, q = 2-, the D.E. (C.i h;is constant coefficients, 

 and the solution is an algebraical function of z, as pointed out 

 by H. A. Webb, in k Engineering,' Nov. 1917. 



With q=l, the solution of (C) is y — C p +i(z) 9 as in Kirch- 

 hofFs solution of the vibration of a conical tapering rod. 



With p = J, 7 = 1, a parabolic section in plan, and tapering 

 uniformly in depth, n = f, 



y = A r-co^(V^ + g) , sin (2^ + 6) 1 



{ -gj- oKyg + iy) + Bh(V^1 ^ ^ (7) 



with the four arbitrary constants A, B, e, 7; : but e and % 

 must vanish if 3/ is to be finite at the point : = 0. 



Generally, if the rod is of circular or square cross section, 

 p = g\ the D.E. (B) cannot bo reduced immediately to the 



