ON THE SPECIAL PROBLEMS OF DYNAMICS. 221 
aX, yX, zX=90°—Z, 90°—Z', 90°—2", x 
Z° Y¥Xa, YXy, YXz=n, 7’, 7", 
then the formul of transformation are 
x ¥ Z 
eee 
w |sing |cos{ sinyn |cosf cos7n 
y |sinZ’ |cosZ’ sinn’|cosZ' cos! Zi ¥ 
z | sin 2” | cos Z” sin n"| cos Z" cos n” 
= O§ 
with the following equations connecting the six angles, viz., if 
—A*= cos (9!—7") cos (n!'—n) cos (n—n!), 
then 
—A 7 —A ae 
OT 1 a Re = tam ofa 
= e05 (n'—n") + o08 (1"—n) : cos (7—7') 
133. It is right to notice that these values of £, Z', £2” give the twelve 
equations a*+2°+ y°=1, &c., but they do not give definitely a=p'y"'—B'y', 
&e., but only c= +(B'y’—f"y’); that is, in the formule in question the two 
sets of axes are not of necessity displacements the one of the other. In the 
same memoir Euler considers two sets of rectangular axes, and assuming that 
they are displacements the one of the other (this assumption is not made as 
explicitly as it should have been), he remarks that the one set may be made 
to coincide with the other set by means of a finite rotation about a certain 
axis (which may conveniently be termed the Resultant Axis), This considera- 
tion leads him to an equation which ought to be satisfied by the coefficients 
of transformation, but which he is not able to verify by means of the fore- 
going expressions in terms of @, Z', 2", n, n', n". 
134. I remark that Euler’s equation in fact is 
a—l, B »Y =0, 
@ 6, p'—l,y' 
al’ : " F y'—1 
or, as it may be written, 
4 i B 7 —(B'y"—B"y')—(y"a—ya")—(aB'—a'B)+a+B'+y"—1=0, 
a ; 
in which form it is an immediate consequence of the equations 
a 5 B : Y, —ie a='y"'—Bp'y', &e., 
ats oe y" 
which are true for the proper, but not for the improper transformation. 
135. In the undated addition to the memoir, Euler states the theorem of 
the resultant axis as follows :—Theorema. Quomodocunque sphera circa 
centrum suum convertatur, semper assignari potest diameter cujus directio in 
situ translato conyeniat cum situ originali;” and he again endeayours to ob- 
tain a verification of the foregoing analytical theorem. 
136. The theory of the Resultant Axis was further developed by Euler in 
the memoir “ Nova Methodus Motum &c.” (1775), and by Lexell in the me- 
