DIFFERENTIAL EQUATIONS OF DYNAMICS, ETC. 
327 
referred to the axis of x, and (3 for the angle between the axis of a? and the node, the 
usual formulse of rotation give 
/'=&/„ cos sin /3 ') 
sin <=<^0 sin jS+fWi cos/3 /■ (38.) 
^’■= 0 )^ — / cos / J 
If in these expressions we put <^2=0, and call 3^ the angle between 
the radius vector and the node, so that <WoCoS|8— (Wi sin(3=rcosS^, ft)oSin/34-*', cos^ 
=rsin^, we obtain finally 
/=acos^, /3'= — a sin ^ cot /, 
sin » ' 
in which the expression above given for a is to be substituted. 
The actual value of is which (since r^, &c. are supposed 
given in terms of t) may be expressed in terms of the four elements first mentioned, 
and t. 
I propose to consider the transformation of the differential equations of the planetary 
theory in a more general manner in the following section. At present I shall add 
some remarks on normal transformations in general. 
74. Theorem. If 
dZ 
dZ 
dy’ dx, 
(89.) 
be a system of 2n simultaneous differential equations, where 
Z=/(^i,3/„ 3^2, 3/2, .-•p,q,r, 
a,ndp,q,r,... are also explicit functions of Xi,&c.,pi,&cc. and t, but are exempt from diff’er- 
entiation in taking the differential coefficients and if these equations be trans- 
(IX I ^yi 
formed by a normal substitution of new variables &c., &c. (art. 62, equation 
(73.)), then the transformed equations are, as in art. 64, 
yJ_d^j^dZ 
dfii dtii 
<_ dZ 
dli d.^i 
in which Z is expressed in terms of &c., 1 ^ 1 , &c., but the differentiations with respect 
to I;, 7ii are performed before the substitution of these variables in/?, q, r, &c.; in other 
words, p, q^r , ... are still to be exempt from differentiation in forming the differential 
equations. 
This may be proved simply by repeating the reasoning of art. 64. The only dif- 
ference is, that in the term 
J <^Z dxj , dZ dyj 
^\dxj dy^i~^ dt/j drii 
the differential coefficients ^ are now taken only so far as Z contains Xj,pj inde- 
