388 Prof. C. T. Knipp and Mr. L. A. Welo on a 



terms of the series, 



M . C*r dx /1\ 2 1 (x-r) 2 dx 



_ /l . 3\ 2 1 1 (x-rydx / 1.3.5 \'l 1 (0— r) 6 <fa "1 

 \2.4/ 3a 4 a 2 -(>-r) 2 V2.4.6/ 5 a 6 a 2 -(^~r) 2 J 



C ri T xdx (1V1 x(x-r) 2 dx 



~ 27ran ) la?-{x-ry-\2) *a*-(m-rf 



_ /l . 3\ 2 1 1 a(a?--r) 4 <fa? / l . 3 . 5 \ 2 1 1 x{x-r)Wx ~\ 

 \JTl) 3a'a 2 -(x-r) 2 \2 . 4 . 6/ 5^ 6 a 2 -(^-r) 2 J- 



Introducing a new variable y=a? — r, we have, after col- 

 lecting terms wherever possible, 



+ W a 2 a 2 -?/ 2 V2 . 4/ 3a 4 1 : ] a 2 -f 



/1.3\ 2 1 1 f dy / 1.3.5 V1 1 _ rt 



+ V2 . 47 3 a 4 a 2 - ?/ 2 \2 . 4 . 6/ 5 a 6 ^ rj a 2 -y 2 

 / 1.3.5 \ 2 11 y 7 4y 1 



+ \2.4.6J 5a 6 a 2 -t/ 2 -T 



Integrating, collecting terms, restoring the variable x and 

 putting in the limits, we have, finally, from the four terms of 

 the series with which we started, 



M = 2iranf^ 



( 2a 



["l - ^Y - /i^- 3 YI - / 1-3.5 \»11 w (a + *i-r)(a + r) 

 L \2J \2.4/3 \2A.6J 5J ge (a-^ + r)(a-r) 



_1[~1 ^Y /1-3V1_ A.3.5 \ 2 1-| 1 a 2 -r 2 



2L V2J V2.4/3 V2.4.6/ 5j l0ge a 2 -(^-r) 2 



_ I |7iY 4- f lif?Yi / 1.3.5 \ 2 ll gl ( gl -2r) 

 a 2 L\2/ i "V2.4/ 3^2.4.6/ 5 J 2 

 £i_r p / 1^3x2 X / 1.3.5 \ 2 ll fo-ry + r 8 

 + a 4 L\2.4/ 3 + V2.4.6/ 5J 3 



I[Y L1Y X 4- fl-3-5y 11 ( ^-r) 4 -r* 

 a 4 L\2.4; 3 + V2.4.6/ 5J 4 



*!-r f/^l . 3 - 5\ 2 ll fa — r) 5 + r* 



- mm] 



1 r/ 1.3.5 Yl-](«-r)'-^> ... 



^USTO^sJ 6 J ( 6 ) 



