Broad River flowing down an Inclined Plane Bed. 27 3 



43. We have not now any such simple partial solution as 

 that of §§ 34, 35, 36 for the sub-case there dealt with ; and 

 we proceed at once to the virtually inclusive* investigation 

 specified in § 37, and, as in § 38, assume 



V — fW+mx+qz) ty (63), 



This gives 



It = la ' i= lm > and V 2 =^-« 2 -? 2 . (64): 

 and (62) becomes therefore 



or, for brevity, 



d*Q) d 2 Q) 



»^ + (e+fy+gf)~+(h + ky + lf)V=0 . (66). 



To integrate this, assume 



qj=c + c 1 y + c 2 y 2 + c$ d + c 4 y* + &c. . . . (67); 



and, by equating to zero the coefficient of y l in (Q6), we 

 find 



(•' + 4)(i + 3)( 2 * + 2)(i + l> C . +4 + (i + 2)(t + l)ec i+2 



+ (»>l>A + i + [*X*-l)fl f +-*>i+*»*-i + foi-. BB • ( 68 )- 



Making now successively ^ = 0, i=l, i = 2, . . ., and re- 

 membering that c with any negative suffix is zero, we find 



4.3.2.1. fiCi + 2 . 1 . ec 2 + 7ic =0, 



5 A .3 .2 . fjuc 5 + 3 .2 . ec 3 + 2 .1 .fc 2 + hc l + kc =0, 



6.5.4.3./xc 6 4-4.3.^ 4 + 3.2./c 3 +(2.1.^ + A> 2 + ^ 1 + ^ =0, >( 



7.6.5.4.^c 7 + 5.4.605 + 4.3./c 4 + (3.2.^ + A)c 3 + ^2 + /c 1 = 0, 



&c. &c. &c. 



These equations, taken in order, give successively c 4 , c s , c 6 , . . ., 

 each explicitly as a linear function of c , c 1? c 2 , c 3 ; and by 



* The Fourier-Sturin-Liouville analysis (Fourier, Theorie de la Chaleur ; 

 Sturm and Liouville, Liouville's Journal for the year 1836, and Lord 

 Rayleigh's ' Theory of Sound,' § 142, vol. ii. shows how to express an 

 arbitrary function of .v, y, z by summation of the type solutions of §§ 37, 

 39 above and § 43 (63), (67), (70) here, and so to complete, whether for 

 our present case or former sub-case, the fulfilment of the conditions (26), 

 (27), without using the method of §§ 34, 35, 36. 



Phil. Mag. S. 5. Vol. 24. No. 148. Sept 1887. T 



