110 
PROFESSOR DONKIN ON THE 
and we shall have (art. 19.) 
V=k(4> cos cos j) -\-^udQ ; 
and we will take 
(46.) 
h, cos i, cos j 
for normal elements *, so that k is to be considered as a function of these elements, 
given by the equation (see (44.)) 
Zr 2 = 
2AC h 
C-(C-A) cos s > 
(47.) 
It is to be observed, that, according to the hypotheses admitted above, k is positive ; 
also, the expression for u at the end of art. 40 becomes, in the case now considered, 
u— Ad'. Thus u has the same sign as & ; and since 6 is evidently comprised between 
i—j and i+j, if we suppose i and j both acute (as in the figure), sin 6 is always posi- 
tive ; hence in the expression (45.) for it, we have to attribute the sign + or — to 
the radical, according as 6 is increasing or diminishing, or according as 0 is between 
o and 7T, or not ; thus, in the position represented in the figure, the negative sign 
must be taken. 
43. If we put +Q for the radical in question, the expression for udO is easily trans- 
formed into the following, namely, 
7 . . k sin f , 1 (cos j— cos z) 2 1 (cos; 4- cos z) 2 ] 
«<*=±-q-{ 1-2 — i-cost ~2 1 + co.r }> 
in which it is evident that the part within brackets is positive upon the whole, but 
each of the two last terms is essentially negative. The integration is now easily per- 
formed, and the result is 
^udd—^rkV, 
where the sign is that which belongs to the radical Q, and P is given by the equation 
„ , cos 6 — cos z cos ; 
sin z sin_; 
(cos z— cos z) 2 , 
r — 1 + COS Z COS 1 
1 . .. . 1 — COS J 
— ^(cos J— cos*) cos 1 
sin z sm; 
2 
(cosy+ cosz ) 2 j 
, 1 / • . *\ 1 + cos 0 
-{-^(cosj-j- cos*) cos 1 
cos z cosy 
sin z siny 
+ an arbitrary function of i,j, h. 
This apparently complicated expression has a very simple geometrical signification ; 
* Since g, h, k are normal elements (i. e. satisfy the conditions [y, A] =0, [A, A:] = 0, [k, g] = 0, art. 31 .), any 
three independent combinations of them are obviously normal also. 
