5)4 
MR. DAVIES’S GEOMETRICAL INVESTIGATIONS 
certain limits; but our equations give no intelligence of their existence or character 
under the aspect we have yet viewed them. 
d y % t 
The general expression of ff given in (85.) combined with (83.), viz. 
d y sin 3 0 ; — sin 3 
dx sin 3 6, cos + sin 3 Q n cos Q n 
with cos 0 I + cos D u = 2 cos (o 
is not in a rational form , and therefore does in itself for all corresponding values of 
• d y 
cos 6 t and cos 0 n involve multiple values of tangent of inclination, There are, in- 
deed, for each value of cos 6 t four points in the curve, all whose tangents ought to be 
d. y 
included in the expression of the values of -j-. These four values are 
dy _ ± { v'p — cos 2 df + V { 1 — (2 cos /3 — cos 0 ; ) 2 } 3 | 
d X ( nnc (\ — nnc — JL c 1 n ~ & SeC 
and 
d x 
(cos d, — cos /3 Y — 3 sin 2 /3 
±{^(I = cos 3 fl ( ) 3 — { 1 — (2 cos /3 — cos 
sec (3 
(99.) 
(cos 6, — cos /3) 2 — ^ sin 2 /3 
This method of proceeding, therefore, has the advantage (and in all cases where it is 
applied the same is true) of giving directly the several inclinations of the tangent to 
the axis of x or y corresponding to any assumed value of 0 t : but for the complete and 
certain determination of the multiple points of the system, other and additional con- 
siderations, as stated in the prefatory remarks to this paper, are necessary. At pre- 
sent we may pass the subject over, as the only remaining multiple points have already 
been determined, as well as the conditions of their existence, in (XXII. q.), and 
proved at the beginning of (XXIV.). 
Finally. It was stated at the close of (XXV.), that when the curves themselves were 
the final ones, or coincided respectively with the magnetic axis, the tangent atM was 
indeterminate ; and the statement is thus rendered evident. 
When the curve coincides with the axis cos 8 = 0, and the equation of the curve 
becomes cos -f- cos 0 n = 0, or = v — 6 n . Hence sin 6 t sin 0 IP and cos 6, = — cos 6 ir 
These values of and 0 tl inserted in (85.) give 
<hi sin 3 6, - sin 3 fl , 0^ 
d x sin 3 0 t cos 0, — sin 2 fl^os 0, 0 ^ 
Moreover, if we take the successive differential coefficients to infinity, (as the value 
of n,) we still evidently have 
d n y 
0_ 
O’* 
Hence there is no factor in (100.) which is determinate; or the inclination of the 
tangent to the axis is at that point essentially indeterminate. 
