INVERSE METHOD OF DEFINITE INTEGRALS. 335 



But when Q„, q„ are maxima and minima, -^ and -^ respectively 



vanish ; therefore, between the corresponding limits of (p, we must have 

 U Q» = 0, f^qn = 0, which general property is easily verified when 



Q„ = «cos«0 and q„ = a sin {n + 1) (p. 

 16. To find the complete integral of the differential equation 

 tt' , -^-^ + (m + 1) (1 - 2t) ^ + n (n + 2m + 1) tr =s 0, 

 where n is integer and m any constant. 



The differential equation for Q„ (Art. 15.) is of the same form as 

 the above equation, and therefore u=cQ„ is a particular solution, c 

 being an arbitrary constant. 



The form of the differential equation for q„ will become the same 

 as that of the given equation, if — (w + 1) be written instead of n in 

 the former; hence, another particular solution is c'q-^„+^y 



The complete solution is therefore 



u = cQ„ + c'q.^„+iy 



This solution fails first when m = 0, for then the functions Q„, 

 5'-(«+i) in their expanded forms become both identical with Laplace's func- 

 tion P„, and consequently the two constants c, c' merge into only one, 

 viz. their sum ; but if we put generally 



b , , b 



c = a -\ — and c = , 



m m 



then M=«Q„ + &. ^"~ ^-'"^' 



m 



And putting m = 0, the latter term becomes a vanishing fraction, and 

 therefore, 



u = aP, + ^^{Qn- S'-(«4.i)} when m =0. 



