324 
PROFESSOR DONKIN ON THE 
Introducing the above value of omitting the brackets, we obtain for the 
system of transformed equations, 
I dZ I / dZ I 
— — co^z, u— — -j — j-iyaV — 
OjUj (XJU 
f dZ I I dZ I 
y' = —-\-00QZ — u^X, v= — ~-\-Ci)aW — 
dy 
dv 
I dZ I I dZ I 
Z=-. \-a}^X — COM, w= --\-co{U — UM 
dw 
( 86 .) 
in which Z is supposed to be expressed in terms of the new variables. 
73. On the principles of the integration of this, and of transformed systems in 
general, I shall make some remarks hereafter. For the present, the following may 
be observed. If, in the transformation of the last article, we suppose the motion of 
the new axes given, then Ao, &c., and therefore also are given explicit func- 
tions of t. But if the motion of the new axes is only given by connecting it with the 
motion of the point m itself, then the above quantities are given functions of the 
variables and their differential coefficients. 
The most interesting case of the latter kind is that in which the motion of the new 
axes is assumed to satisfy the equations 
Wq <«[ Wo / X 
- = - = -% 
X y z 
which express the condition that the instantaneous axis of rotation (of the moving 
axes) always coincides with the radius vector of the moving point m *. 
The radius vector traces, in fixed space, a certain conical surface. It also traces, 
with reference to the moving axes, another conical surface ; and we might always 
assume as one of the conditions defining their motion, that this latter should be any 
proposed surface ; that is, we might assume that the new coordinates x, y, z should 
always satisfy the equation <p{x,y, «)= 0 , (p representing any given homogeneous func- 
tion. If to this last assumption we add the two conditions expressed by the formula 
(w.), we further assume that the conical surface traced hy the radius vector with refer- 
ence to the movhig axes, rolls upon that traced infixed space. 
Suppose, for example, we assume for the equation (p{x,y,x) = 0, simply 2=0. This, 
with the conditions (w.), will express that the radius vector is always in the plane of xy, 
and that this plane rolls upon the conical surface traced by the radius vector in fixed 
space. We may then say that the plane of xy is the “ plane of the orbit,” and that 
the axes of xy, or any lines fixed with reference to them in their plane, are “fixed in 
the plane of the orbit'f'.” 
* See Jacobi’s first letter to Professor Hansen (Crelle’s Journal, vol. xlii. p. 21). This letter appears to 
refer to some unpublished {}) results of Professor Hansen, which may possibly be similar to those of this 
article. 
t The student of elementary treatises is, I believe, always left to find out for himself what this expression 
means, or ought to mean. 
