526 PROFESSOR CHRYSTAL ON 
This affords us convenient means for exploring the complex by cna 
radii through the centre. 
We shall in the first place investigate the rays passing through any radius 
and perpendicular to it. 
If p be the distance of any point on the radius (/mn) from the centre, the 
direction cosines of the two rays are given by 
(Pf )2+ (p—g?u2+(2—H*=0, (17) 
Dd + mp + nv =" (18) 
The most important question is whether the pair of rays is real for all 
points of the radius. If /? g? h? be all positive and different from zero, it is 
clear that no ray can pass through the centre; and, if they are all finite, no 
ray can be at an infinite distance from the centre. 
In other words, there must in general be maxima and minima values of p. 
The finding of such when the conditions are as in (17) and (18) leads, as is well 
known, to the following results— 
12 m2 n? 
pape + pag t 
=) ; : : # (19) 
Eee ua ie 
MELE: age aetna . > (20) 
(19) gives the maxima and minima values of p?, and (20) gives the direc- 
tion cosines of the corresponding rays, which, as may be easily verified, are at 
right angles to each other. 
It is easy to see, moreover, that the roots of (19) are always real and 
positive, for if /? g* h? be in order of magnitude they clearly include the roots 
in their intervals. 
Another method leads to interesting results. 
If 6 be the angle between the rays whose direction cosines are given by 
(17) and (18) when p is assigned, then we get very easily 
— 2) = {Pp — 9) (p?— hi) + 2p? —h*) (? —f*) + 020? —f) (0? —9") 
tan = GDP G+ Emap I) 
Hence if p,? and p,” be the roots of the equation (19), in order that @ may be 
areal angle p* must be between p,” and p,?. When p*=p,? or =p,”, 02=0; and 
when p?=} (p,2+p,2), => x: The last of these results is a particular case of 
