Mr. R. Pendlebury on some Definite Integrals. 4-39 



2 J, 



fifo 



^C 1 -^)- • • • • (8) 



It is clear now that, by the various transformations to which 



C 1 e~ a2x2 dx 

 the integral I g can be subjected, we have a new series 



of definite integrals opened out, which may perhaps be worth 

 the trouble of arrangement and tabulation. a 



The fourth power of the Error-function can be easily ex- 

 pressed as a single integral by the method adopted to determine 

 the value of the square. 



We easily get from (7), 



J, Ji l+* 2 + 2/ 2 + *V 



S 



.d P dd 



+ p 2 +p 4 sin 2 0cos 2 



1+tf 



The limits are 



= i e -2a2 ft £f_ _ arc tan ... * . tan 26 do. 



5 are 



/;\ /)_ arc tan Vp«-l"l _ i/jH 



(ii) " = arc cot V^l} P= l }' 



-I 



2 ^ oo 



=oJ "= 



(iii) *=*r p=v^j- 



The last of these three terms vanishes. The others are equal* 

 and give 



