H. E. SoPER, A. W. Young, B. M. Cave, A. Lee, K. Pearson 331 
and we have 
2% — 1 n — l , 
Here 
pr Vl - p2 Vl - r2 __ (1 - jo^) (1 
1 _ p2,^.2 ' '^2 - 1 _ p2^2 
are constant for p and r given and thus (viii) enables us to deduce ?/„+2 fi'om y^+i 
and ?/„ for a given p and r. But by simple differentiation 
1 — p^ f 1 pr cos~^ (— pr)^ 
77 
2/4 = 
1 - / 3/3r (1 + 2ph'^) cos-1 (- pr] 
77 
(,T 
.(ix) 
p2r2)2 _ ^2^2^| 
Hence if ^/g and be calculated for a series of values of r and p all higher values 
may be reached by a repeated use of (viii). The values chosen were : p proceeding 
by •! from 0 to 1 and r proceeding by -05 from — 1 to + 1. 
The disadvantage of this method of calculating is that, except by inde- 
pendent computing, there is no means of checking accuracy until all the ordinates 
have been deduced, and any mistake in y„ for a low value of n is perpetuated 
throughout the series. When all the ordinates have been found, say for n = 25, 
then the smoothness of these ordinates and the fact that they give the correct 
total area with a suitable graduation-formula will be checks on the accuracy of 
the whole system of ordinates. In this manner Table A, p. 379, was calculated. 
Another method of approaching the value of y^ is of some advantage. We 
may take 
n-l 
_ (1 - P") ^ n ,^2>|^ / . % CO S-1 (- pr) \ 
(1 - pV) 2 
where v„ and are functions of pr in integer positive powers, and if we substitute 
in (viii) we obtain 
«n+2 = i^n - l)prv„+i + (n - 1)^ (1 - p^r^) v„ 
Un+2 = (2W - 1) prUn+i + {n- 1 f (1 - p2r2) Un] 
We may write y^ in the form 
n-\ 
(l-p2)~r (cos-M-pr) 
"~ C/v, o \ t _ ^ 
(xi). 
2) ! 77 ' ' d [pry-^ \ Vl 
p'-r 
or V - in- 2) ( ^ + co s-^ {- pr) \ 
