MULTIPLICATION OF ELLIPTIC FUNCTIONS. 903 



Xow, q being equal to 



The complementury function is thus 



but, as ^fi(ii — ^) is odd and H^_^ contains only even powers of ç, C" = 0. 

 Ao-ain, a jmrticular integral is -2^'(?i-l>?.$^"<"-i>-2-2^-^;?(» + ])ç^"("-i^+-', 

 which, together with the complenienturv function just found, gives 



where P^ is an as-yet undetermined function of n. 

 When —^ is odd, we find likewise 



r>y virtue of (HI), 



E^_,= r, e^"('^ ;'.)-i_2p-=';,(,i + l)f^"("-i)-^_'2"-«^;?,- l)»f^"(''+i>+i, ^i^ even, 



^^_i=2"-\'n-l)».?*"<"-i^-='+2^-^';K'«+l)?-"^"~^^+^-^i çi"'"+3>-\ '-^ odd. 

 We may put the multiplicator n in evidence by writing 



and then e(ju;ition (112) may be written 



whence, equating the coefficients of the second highest power of a, 



»1+2 n—2 )i— 2 »1+2 r~ n —12 p « -12 



»I n 



