834 Mr MURPHY'S THIRD MEMOIR ON THE 



then assigning to m all possible values from — oc to +00, the functions 

 Q». qn will give an infinite series of reciprocal functions relative to all 

 the transcendants contained in ^ considered as the variable of integra- 

 tion ; and when m is between - 1 and + 1, pairs of reciprocal functions 

 will be obtained, except when ?» = 0, when both coincide. 



In this series are included the trigonometrical functions, namely, 

 when m-= —\'., and Laplace's functions, when /« = 0. 



In all the reciprocal functions thus arising, there exists one common 

 property, namely, the definite integral always vanishes between the 

 limits which make the functions themselves maxima and minima; this 

 remarkable property I have had occasion in another place to notice, in 

 the particular case of Laplace's functions.* 



To prove this generally take the equations of the preceding article, 

 viz. 



/#'^ + (»» + l)(l-20.-^'' + «(w + 2»« + l)Q„ = 0, 



«'^ + (?» + l)(l-20•-^+(« + l)(w-2»^)9„ = O. 



Multiply both equations by {tiy, and integrate reserving the con- 

 stants under the integral sign ; hence, 



{tt'Y^^ ^ + « (« + 2m + 1) j: Q„ (tty = 0, 



' (^0'""''-^+ (« + l)(«-2»^)/,^„ («')"* = 0; 



and changing the independant variable by the condition 7;r =(^^')"'"» we 

 have 



(«')""+^^ + w (« + 2/» + 1) 4 Q„ = 0, 



(<0*"*' ^ + (» + 1) (»« - 2»w) /^ ^„ = 0. 



Electricity, Introduction. 



