ON THE SPECIAL PROBLEMS OF DYNAMICS. 219 
And he remarks that this is in fact the problem of the transformation of coor- 
dinates, viz., if we have 
X=ax + By + 2; 
Y=c'r +B'y +y'z, 
7, =a"e+B"y+ yz; 
then the first equations are such as to give identically 
XP 4 Y?4 2247+ y?+427, 
126. Assuming the first six equations, he shows by a direct analytical 
process that a*=(B'y"—f"y')’, or z= +(6'y'—B"y'); or taking the positive 
sign (for, as the numbers may be taken as well positively as negatively, there 
is nothing lost by doing so) a='y'"—" y', which gives the system 
Zz ey ey,» B =y & ya to =o) 0 8, 
a =f Y me M4 2 py Ke pr: e 2 aes B xe B , 
a=By—By, B'=ye—yYa, y"=a B'—a' fp, 
and from these he deduces the second system of six equations. The inverse 
system of equations 
X=ar+a'y+a"z, 
Y=6er+ p'y+ B"z, 
Z=yetyyty"2 
is not explicitly referred to. 
127. He then satisfies the equations by means of trigonometrical substitu- 
tions, viz., assuming a=cosé, then @?+a@'*=sin?Z, which is satisfied by 
a =sin { cosy, a =sin £ sin n, &c., and he thus obtains for the coefficients a 
set of values involving the angles Z, n, 0, which are the same as those men- 
tioned post, No. 130. And he shows how these formule may be obtained geo- 
metrically by three successive transformations of two coordinates only. The 
remainder of the memoir relates to the analogous problem of the transforma- 
tion of four or more coordinates. 
128. I have analysed so much of Euler’s memoir in order to show that it 
contains nearly the whole of the ordinary theory of the transformation of 
coordinates ; the only addition required is the equation 
=+1, 
where the sign + gives a=('y—"y’, &c. (ut supra), but the sign — would 
give a=—(P'y"—f"y'), &e. 
129. The distinction of the ambiguous sign is in fact the above-mentioned 
one of the proper and improper transformations ; viz., for the sign + the two 
sets of axes can, for the sign — they cannot, be brought into coincidence: 
this very important remark was, I believe, first made by Jacobi in one of his 
early memoirs in Crelle’s Journal, but I have lost the reference. As already 
mentioned, it is allowable to attend only to the proper transformation, and 
to consider the value of the determinant as being =+1; and this is in fact 
almost always done. 
130. Euler’s formule involving the three angles are those which are ordi-- 
narily made use of in the problem of rotation and the problems of physical 
astronomy generally. 
It is convenient to take them as in the figure, viz., 0, the longitude of node, 
