510 Sir Gr. Greenhill on the 



In tbe more general case o£ (4) § o, as 



*»§+(* +■**)*.= <>, .... (5) 



first change the dependent variable, by y=z 1 w, 



a cPy ,f Ai dw - "I 



*» JT + 2?* g + [q(q - 1) + A + fo*>] « = 0, . (7) 



of the for 



in 



d 2 w die , 7 „. ,, /m 



-^ + as -- + ( c + ***> = ; (o) 



and then, taking 9(9— 1) = —A, 2# — 1= \/(l — 4A), 



^^+2}^ hfe*w=0 (9) 



Next, with a new independent variable x = kzv[p 2 , 



die die 'd?w dw 9 / „d 2 w . rfzpN /1A , 



z di = **&• 2 s + -mi = p r s? + *a> • (10) 



« t i ,w V p / «•» 



/kzP\ 2q — 1 



Clifford's form (5) § 1, for y = Q n {%) = C 7i i —^ I, rc= — — . 



To utilise the elaborate tables of the Bessel and Neumann 

 function, computed in Reports of the British Association,. 

 say of argument s, all that is required is a new column 

 of x — 2^/z^ as argument x of the corresponding Clifford 

 function, if the order is zero ; otherwise a factor is required, 

 the appropriate power of z. 



The series of Glaisber's examples in Forsyth's ' Differential 

 Equations,' Chapter Y and elsewhere, w T ith solution in finite 

 terms, are seen in this way to be exercises in the differentiation 

 of the Clifford function of order half an odd integer, if the 

 former variable x is replaced by 2 s/ z. 



And looking back at equation (3) § 1, with y = a? n O»(#) as 

 the dependent variable, the independent solutions are seen 

 to be given of the differential equation (Lommel, Math. 

 Ann, ii.) 



d 2n n 



*fiJs=y**yy="*Ur*) or *-i>.(««), •»=!. (is) 



