320 Ma MURPHY'S THIRD MEMOIR ON THE 



The first term, and 'a fortiori', all the succeeding terms of this 

 series vanish between the limits ^=0, and t=\, or t' = 0, for 



d''-'{{tt'rV} _rr d"-'{tt'Y ,,^ ^,dV (f-^tty 

 dt--' ~ dt^-' ^^ ' dt dt"-' 



{n-l){n~2) dT d^-^itt'f , 

 "^ 1.2 dt' ■ dt"-' "^ ' 



the first term of this latter series contains a factor tt', the second a 

 factor {tt'f, &iC., and therefore the whole vanishes between limits. 



The following exception to this theorem must however be attended 

 to; V must not he of the form {tt')".Vj, where r is equal to, or 

 greater than unity, for the above reasoning will not be applicable, 

 since then 



d"-mtt'Yr\ _d'-'{{tty-^r,} 



_i 



dt"-^ dt' 



which being expanded as above, will not vanish unless r be less than 

 unity. 



4i. If a function V can he determined so that the quantity 



d''f(ttrvi d0 



at" ■ dt 



may he of n dimensions in t, then the factor hy which -^ is here 



multiplied, will he a self-reciprocal function when the integrations are 

 performed relative to cp. 



Denote this coefficient by q„, then 



r -r ^0 _ /■ <^" (tf'Y ^ d<t> 



and as we may suppose m<n, the general term of qm-^, as a^t^ 



cannot be of greater dimensions than n — 1, and therefore the part of 

 the whole integral dependant on this term vanishes, as has been 

 shewn in the preceding article, hence f^qmq„ = 0, when m and n are 

 unequal. 



