Manchester Memoirs, Vol. Ixvi. (1922), No. 1 3 



necessary first to calculate by quadratlire of the integral (2.3), 

 and subsequently to perform a second quadrature of the 

 integrals (2.1), (2.2), using the computed values of 6. None 

 of these three integrals, however, is in a form suitable for 

 numerical calculation, since in (2.3) the integrand is finite at 

 the upper limit, while in the other two the range is infinite. 

 It is therefore convenient to transform the variables so that 

 these infinities shall disappear, leaving the integrals in a form 

 to which Simpson's or Weddle's rule can be applied. 



In the case of (2.3) it is desirable to take y]o rather than a 

 as known, since it is difficult to calculate y]o from a, but easy to 

 find a when y]o is given. Thus, if y]o satisfies (2.4), we have 



while 



<3-») -.■-^(j)-=.---^(jr(-:r 



=(i-„2)-(i-,„^)(^Y ' 



Now introduce new variables ;x, Xo, defined as follows; 



(3.3) ri = sinx rio = sinxo> 



Xo being a function of a according to the relation (cf. 3.1) 



(3.4) a^ = sin^y^ , f — ^r- F^ 



w 



hile 



(3-5) J -r]2- -^ y^ -cos\-{sinyJsinyoY-'' cos^o- 



Expressed in terms of 7, (2.3) becomes 



'{ i^{sinylsinx,y-^ {cosy,lcosyy)''"dx 



the limits being o and -/o- Finally we transform to the variable 

 Zy where 



(3.7) 02^ I -X, y=^^ro^ 



so that 2 ranges from o to i as \ varies from i to o. The 

 expression for 9 then becomes 



(3.8) - 27 



;^ (i-{sin\y,lsiny^Y-^(cosyJcos\yoY^^ -' , 







../■ 



=,2x0 ct>dz = 2/^xo 



