314 Mr. R. B. Sangster on some Consequences of 

 Substituting the volume formulae and writing ^S for S', 



n 2 IVx + V ^ U+%7 J 



In expressing the values of % in this equation some abbre- 

 viation will be an advantage. 



Let 2n{^ 2 -iy-?i 2 ({M 4 -6n 2 + l)=p, then 



£ + l±i/_P = _A 



* yu, 2 (n— 1) A /* 2 ' 



The next step is to add the square of half the coefficient of 

 ^ to both sides of the equation, and, for future reference, 

 it has to be noted that in making this addition, y/p in the 

 coefficient of % must have the same sign simultaneously 

 attributed to it in both sides. Again abbreviating, let 



(y" 2 + 1 ± \Zp) 2 - VO - 1) 2 = q, 



then _ yu, 2 + 1 + y/p ± K /q 



X = 



2p\n-l) 



Tn dealing with this formula numerically it is generally 

 more convenient to write the equivalent of p and q as 



p - (2n-7 i 2 )( A t 4 + l)- 2^ 2 (2h-3/* 2 ), 



q = (2n - n 2 + 1) (^ + 1) + 2fi\2n + n 2 ~ 1) ±2(> 2 + l)y/p. 



It will be noticed that-^/jt? occurs twice in the values of %, 

 but the origin of the double occurrence, in the process of 

 extracting the roots, demands that a similar sign, either + 

 or — , shall be attached to the two occurrences. There are, 

 therefore, four roots to the equation. 



We have one pair of values of % in which s/ q is positive ; 

 these are applicable to the ca?e of /x>l. The pair in which 

 y/q is negative are for use when /x is < 1. In order to show 

 this, write fi = l in the values of % and in the sine formula 

 (4). When — y/p is adopted, the result is %= —1, which is 

 an impossible value of %, since % must always be positive, 

 and the reason for this result will be seen presently. When, 

 however^ we adopt + v//?, then -\- yf q makes 



% =( v /»+i)/( v /«-i), 



and —y/q gives % — (sjn — \)\{y/n + 1), and in both cases 

 the resulting angle is 90°. Now it is obvious that as i 

 approaches indefinitely near to 20°, S'/S=% is > or < 

 unity according as fi is > or < unity. But -\-y/q gives 



