CONCERNING TERRESTRIAL MAGNETISM. 
93 
nite sphere diametrically opposite to M. These results accord with the statements 
made in article XXII., and justify them. 
To find the values of ^ corresponding to these multiple points, we may proceed 
thus : 
When sin 6, = 0, we have cos 6 t = + 1 ; and cos 6 U = 2 cos (3 + 1, the lower sign of 
which being impossible, so long as cos (3 is positive (which is our hypothesis) in these 
investigations ; and though we might have so extended them as to include negative va- 
lues of cos (3, yet nothing in point of generality would have been gained thereby, as 
only a reduplication of the branches of the curve would have resulted from it), and 
hence the only solution is cos 0 U = 2 cos /3 — 1. But this final direction of r u is the 
direction of the tangent at U : and as the same holds for cos (— 0J, there are two 
tangents at U equally inclined to the axis. 
The value here given accords with the construction given (in XXII. m.). For 
(figg. 10, 12, 13.), taking P U as radius = 1, UH = UT — T II = 2 cos (3 — 1 ; and 
it is the cosine of the angle RUIl by the construction. Hence it fulfils the condition 
of the equation and gives the tangents at the point U. In the same manner that 
construction gives the tangents at T. 
The combination of the third equations of (97-) and (98.) may also be easily shown 
to coincide with the construction given for the asymptotes in (XXII. o.) ; but as the 
truth of that construction has already been proved at the close of XXIV., it is unne- 
cessary to recur to it here. 
But there occurs here a difficulty which is worthy of notice, but which is readily 
shown, however, to be only apparent. We have seen at (XXII. q.) that there is an- 
other doubly symmetrical system of double points possible for values of f 3 between 
X 
2 
it is quite clear that so long as we require only the differential fraction — , we may eleminate the denominators 
z 
by the usual process before we commence the differentiation, even though they involve functions of the variable 
quantities that enter into an investigation. The same is true of factors not fractional : for 
/ xz\ yz(xdz + zdx) — xz(ydz + zdy) 
\yz) ~ y° z 9 
y z * 2 dx — x z 9 dy y dx — x dy 
y‘ l z~ 
y 
-<T> 
It hence follows, that for seeking the second differential coefficient of the curve, we may eliminate by divi- 
sion or multiplication any common factor, integral, or fractional that enters into the numerator and denominator 
of the first differential coefficient, and hence that the process followed in (XXIII.) is legitimate. But when, 
on the contrary, each of the terms (numerator and denominator) of the first differential coefficient is to be 
equated to some other quantity, or to zero (as in finding the multiple points), then all the factors should, by 
the fundamental principles of algebraic equations, be retained in both. Hence the equations of condition (the 
first and second in (97.)) must retain all the factors which the process of first differentiation introduced into 
them. The want of due attention to this principle, simple and obvious as it is, has often led to very incom- 
plete, and sometimes very erroneous, enumerations of the characters of certain curve lines. 
