180 
MR. GRAVES ON A RECTIFICATION OF THE 
is a value of cos 0, which is always less than r (it being recollected that 6 
is a quantity between 1 and — 1) it follows that the particular value of 
cos '* which 1 denote b y cbs ~' v wh s- 
S r 
~ TP 12" 
R 
1 2 .3 9 .. . (2» - l) 2 
>V B V + ‘\ [29] 
1-WR* + SV J 
R 2 + S 2 ’ ” 1.2...2».27l + l.V i /Ra+S 8 
§ 6. In the equation x A+ a/_1b = y, to determine what real values x may 
possess, so that in each case a corresponding value of y may like- 
wise be real. — [Note I.] 
By [20], 
By [28], 
Hence 
j/ = f{(A + V -1 B)f l x} 
/»— 1 . N 
I X = 2 l 7T -f cos 
V'x 2 
V' -1 1 \/x 2 
V 
= f{(A 
4- \/ — 1 B) (2 i 7r + cos ' /~i ~ V' — 1 1 v'x 2 ) 
V & 
} 
or 
f -j~A ^2/7 t + cos 1 + B1 ^/x 2 + V —1 ^2in + cos 1 ~ ^ V'x 2 
or (see [11] ). 
f«£ A (2in + cos 1 pp) + B1 v .i £b [tin + cos 1 p— ] — A 1 v^x 2 J j* 
In this expression the factor 
f{ V -1 [B (2*V + cos -1 JL) _ A 1 \/J 2 ]} 
is always real, as is evident on developing by [7] . 
Hence, that some y may be real, the other factor, viz. 
f <j^A (2 z'tt + cos 1 p^p) +B1 \/x 2 j» 
must also have some real value. 
Hence (see [G] ) some one, at least, of the quantities 
sin-^A (2z'tt + cos ~ 1 pp ) + B1 v'x 2 ^- 
must = 0. 
