98 
PROFESSOR DONKIN ON THE 
Lastly, if this be combined with (iii.), the following expressions are found for u and w : 
m = i{2mr 2 (A + ?) (r ) ) - A: 2 |^ - A 2 - ^ c 2 
w-= z 7 \2nir\h + p(r) ) - k 2 j "+t\k 2 — r - c 2 j * 
(in which it is to be remembered that r*=z 2 +f), and if to these we join the equa- 
tion (ii.), the values of u, v, w are explicitly given in terms of the conjugate variables 
§, 0, z. We have then (art. 19.) 
Y==^(udg-)rvd()-{-wdz) ; 
or, substituting the above values, 
V= c 0+Jj-^+^ 
The term under the integral sign is easily seen to be (as we know a priori it must be) 
a complete differential. It is convenient however to transform it thus. First, we 
have gdg+zdz—rdr ; next, let the latitude of the body (or the angle between r and 
the plane of x, y) be X ; then tan X=-> and 
r 2 
^dz — zd§=r 2 dX, - 2 =sec 2 ^. 
Making these substitutions, the expression for V becomes 
V = cd-\-^—(2mr 2 (h-\-(p(r)) — k 2 )^ -{-^dX(k 2 — c 2 sec 2 7^. 
The integration in the second term cannot be effected till the form of the function 
<p(r) is given : that of the third term may be more conveniently performed after the 
differentiations with respect to c and k, as in the next article. 
29. The remaining integrals* of the problem are (art. 19.) 
dV 
dk 
= a 
d_V_n dV 
5 dc~P’ ^ 
dh 
Performing the operations indicated, and observing that 
d\ l . , f k sin A 
\wW^? 
and 
J 
V' k 2 — c 2 sec 2 A 
sec 2 \d\ 
ilsin - 1 
k 
Vk 2 — c 2 sec 2 A c 
1 . , / c tan A 
-sin -1 ' 
V Vk 2 - 
) 
)> 
* It would perhaps be better to use the term “ integral equations” here, in order to reserve the term “inte- 
gral” for the case of an equation involving only one arbitrary constant (see art. 23.). The equations — =a, &c. 
dk 
become “integrals” in this sense, when for k , c, and h, on the left, are substituted the functions of the varia- 
bles to which they are respectively equal (from (i.), (ii.), (iii.)). An “integral” in this limited meaning is 
what is commonly called a “ first integral,” when the problem is considered as the solution of n differential equa- 
tions of the second order. And any equation obtained by combining “ integrals” so as to eliminate a set of n 
of the variables x lt x it ... x n , y x , ... y n , of which no two are conjugate, corresponds to what is commonly 
called a “ final integral.” 
