ye> Mr. Gwyther on a 



the series becomes 



y ^ 1.2 J 



= (i-EOX. 



= (i + E + E^ + &c. + E^-i)^( - t^Yu,,, 

 and the series is seen to be zero \i p>ty and \i p = t the 

 value of the series is (^ — s)\t\ 

 Similarly the series 

 n{ri - k)- - -(ti - (/ - i)^) -/(// + k)n{n - k)- - -(;^ - (/ - 2)k) + etc. 



= (i + E + E^ + &c. + E^--i)^X - ^ )X 



= Oif/>/. 



= (- i)7! k"', \ip = t. 

 I shall write down the results of the simplifications, in 

 which I have merely treated y as zero and x^-=^f% but not 

 taken the other steps towards finding the values at the limits. 



7/2 -= 2(«^,. + r„ - ^,>"-^(^ + ^)'"^ + ^ + ^n^"(^^' + ^)"A^" ~ ' 



v'=^ - 2{n.{n - i)a, + 2(n - i)c^~ 2(;2 - 2)/4}^"-%r + ^^''-1 

 - {m + i){m + 2){2na^ + c,, - T,b,)x"-\x + cY" 



- {m + i)(;// + 2)((;/ - i)^, - 2^,>"-=^(^ + cY'lcr^'~\ 

 In this it will be noticed that (i) (;r+^)"'+^ is the highest 

 power of (;ir+<:) which appears; (2) that c appears to the de- 

 nominator of every one of these expressions except that for 

 vi ; (3) that the power of r to the denominator is always lower 

 than that of ,r in the numerator, and that thus any expression 



such as -;p can be written ~^(^x + c-c) or ^^^pcosd-c). 



Before we consider the consequences of substituting the 

 values at the limits, we may notice that, as we are taking 

 into account terms which must be neglected unless we 

 profess a minute accuracy, we must not neglect the terms 

 which will arise from other terms than the leading term in 

 the expression for the original wave which was broken up. 



