358 
ON THE DIFFERENTIAL EQUATIONS OF DYNAMICS, ETC. 
(3) is defined by the equation T’=Z- 
dV 
in which (after the explicit differentia- 
tion of P with respect to t), See., y^, &c. are to be expressed in terms of the new 
dV 
variables, y^, &c. are thus expressible by the help of the m equations ^=;j(and the 
^ /c?L d7i\ 
n-m equations ^j= 0 - 
dV 
If &c., do not contain t explicitly, then -^=0, and "P is obtained merely by 
expressing Z in terms of the new variables. 
It may be observed that the whole of the above reasoning would apply to the case 
in which the new variables |i, ... are more in number than the independent vari- 
jp 
ables of the problem (or m>n—r), with this exception ; that the m equations -^=n^, 
together with the r equations obtained by differentiating the equations of condition 
totally with respect to t, would be more than sufficient to express 3 / 1 , ... yn in terms of 
the new variables ; consequently ?/i, &c. might be so expressed in different ways, and 
therefore, although the value of 'SP' obtained by the above rule would certainly be the 
same as that obtained by recurring to the original formula (D.), the form of might 
be different, and therefore the resulting formula erroneous. 
There must doubtless exist some rule for choosing n — m combinations of the 
equations of condition in such away as to lead to the correct forms of y^, ...y^ as 
functions of the new variables ; but I have not at present attempted to investigate it, 
and perhaps it would be hardly worth while. The theorem in the case in which the 
new coordinates are independent, may, I believe, be practically useful. 
ERRATA IN PART I. 
Art. 1. equation (4.), for dx read dxi. 
Art. 10. In paragraph preceding equation (26.) omit the words “ not containing t explicitly.’' 
Art. 18. equation (/3), for y; read y\. 
Art. 19. equation (29.), for hi read bi. 
Art. 24. second line after equation (L.),Zor “ such as h, k ” read “ such as J", g.” 
Art. 30. The expressions equated to h, k, c, and the three terms in the left-hand column of the table of 
elements, should each be multiplied by m. > 
Art. 42. near the end, for “ according as 0 is between o and tt, or not” read “ according as 0 is between 
IT and 27r, or between o and tt.” 
